Mi^Si'iTI,0.';  CALIFORNIA,  SAN  DIEGO 


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THE   ELEMENTS   OF   LOGIC 


THE    ELEMENTS 


LOGIC 


THEORETICAL    AND    PRACTICAL 


JAMES   H.   HYSLOP   Ph.D. 

INSTRUCTOR   IN  LOGIC  PSYCHOLOGY  AND  ETHICS  COLUMBIA    COLLEGE 
NEW   YORK 


NEW  YORK 

CHARLES    SCRIBNER'S    SONS 

1892 


Copyright,  1S92,  by 
CHAKLES  SCKIBNER'S  SONS 


trow  DinccTonr 

"him  !•,,.  ano  nn.  i-  Hi' .Minn  COMPANY 
NEW  YORK 


PEEFACE 


All  who  have  had  anything  to  do  with  Logic  will  recognize, 
without  being  told  it,  the  extent  to  which  I  am  indebted  to 
Jevons  for  both  matter  and  method  in  the  treatment  of  this 
subject.  But  they  will  quite  as  readily  perceive  the  deviations 
from  him,  and  the  additions  which  I  have  made  with  the  hope 
of  improving  upon  his  work.  Jevons  designed  his  "Element- 
ary Lessons "  to  direct  the  student  in  practical  reasoning  and 
correct  thinking  in  professional  vocations.  I  have  intended 
the  present  work  to  serve  the  same  end,  and,  if  possible,  more 
completely  than  Jevons.  For  this  reason  I  have  been  deter- 
mined in  the  development  of  the  subject  by  the  questions  con- 
stantly put  me  by  students.  As  far  as  possible  I  have  endeav- 
ored to  answer  all  questions  likely  to  be  raised  when  framing 
rules  about  the  logical  treatment  of  conceptions  and  proposi- 
tions. It  will  be  apparent,  therefore,  that  the  plan  has  neces- 
sitated many  judicious  omissions  both  of  irrelevant  matter  in 
Jevons  and  others,  and  of  certain  theoretical  discussions  pecul- 
iar to  the  scientific  rather  than  the  practical  aspect  of  Logic. 
With  a  view  to  the  student's  guidance  and  mental  discipline  I 
have  added  a  large  number  of  practical  questions  and  exer- 
cises at  the  close  of  the  book. 

It  is  important  to  remark,  however,  that  I  have  aimed  to 
satisfy  two  wants  at  the  same  time.  I  have  tried  to  produce 
a  work  that  could  be  used  both  for  beginners  and  for  advanced 
students  of  the  subject,  but  not  for  those  who  care  to  go  into 
it  exhaustively.  Beginners  can  be  directed  to  the  definitions 
and  illustrations,  and  the  heavier  matter  omitted,  while  the 
special  questions  developed  at  length  and  in  a  more  technical 
manner  can  be  taken  up  by  the  advanced  student,  or  by  those 
who  are  interested  sufficiently  to  press  further  inquiries.     But 


vi  PREFACE 

only  one  or  two  chapters  are  inserted  which  do  not  have  ref- 
erence to  the  practical  advantages  of  Logic.  The  full  treat- 
ment of  all  subjects  is  designed  to  afford  students  a  better 
guide  than  Jevons  can  possibly  be.  With  this  in  view  I  have 
laid  considerable  stress  upon  the  nature  of  Conceptions,  Prop- 
ositions, and  the  Classification  of  Fallacies.  The  last  subject 
always  gives  students  a  great  deal  of  trouble,  because  they 
find  no  means  of  distinguishing  one  fallacy  from  another  as 
clearly  as  is  desirable.  By  the  manner  in  which  I  have  tried 
to  treat  the  subject  I  hope  I  have  rendered  their  work  easier 
in  this  important  part  of  Logic.  De  Morgan  is  the  only 
author  I  know  who  has  approximated  what  the  subject  of  fal- 
lacies deserves  ;  but  he  leaves  very  much  to  be  desired.  He 
has  not  attempted  either  a  classification  or  a  discussion  of  the 
way  in  which  several  fallacies  may  coincide  at  the  same  time, 
and  differing  only  in  the  point  of  view  from  which  they  are 
regarded. 

In  the  treatment  of  certain  questions  it  has  seemed  fit  to 
venture  upon  some  distinctions  which  may  be  regarded  as 
innovations.  I  have  distinguished  between  two  kinds  of 
"  General  Terms,"  which  I  have  called  "  Mathematical "  and 
"  Logical  Generals,"  for  reasons  that  the  text  must  explain.  In 
pursuance  of  this  distinction  and  the  double  signification  of 
the  term  "  Genus,"  I  have  also  coined  the  term  "  Conferentia," 
as  denoting  the  "  logical "  as  opposed  to  the  "  mathematical 
genus."  A  term  is  required  to  contrast  with  "differentia,"  as 
"  genus  "  contrasts  with  "  species."  Whether  I  am  justified  in 
the  invention  must  be  a  matter  between  me  and  my  critics 
when  they  have  examined  the  work. 

If  I  mistake  not,  I  have  somewhat  modified  the  ordinary 
t  n  , it  incut  of  Induction.  First,  I  have  carefully  distinguished 
between  "inductive  reasoning"  or  " inference,"  and  what  is 
ordinarily  called  "  Induction  "  or  "  Inductive  Method."  Then, 
to  complete  the  thought  involved  in  this  distinction,  I  have 
discussed  "Scientific  Method"  as  somewhat  extra-logical,  and 
included  therein  the  process  of  verification  as  involving,  in 
some  instances  at  least,  more  than  purely  inductive  inference. 


PREFACE  vii 

The  importance  of  this  will  be  apparent  when  we  observe  the 
constant  confusion  by  logicians  of  formal  Induction  as  a  pro- 
cess of  reasoning  with  the  methods  of  discovering  and  verify- 
ing new  knowledge.  Bacon  and  Mill  have  not  sufficiently  dis- 
tinguished them,  and  on  that  account  have  done  much  to  mis- 
represent and  to  disparage  the  importance  of  Deduction.  I 
have  endeavored,  so  far  as  I  am  able,  to  remove  this  defect 
from  the  discussion.  But  of  my  success  I  am  probably  not  a 
competent  judge.  Besides,  the  limits  to  which  I  have  been 
compelled  to  reduce  the  discussion  may  appear  to  make  this 
part  of  the  work  unsatisfactory.  I  am  content,  however,  if  I 
have  given  a  hint  in  the  right  direction. 

Special  acknowledgments  are  due  to  Mr.  Fowler  and  Mr. 
Venn  for  the  valuable  help  which  I  have  received  from  their 
works  on  Logic.  Mr.  Venn's  recent  volume  on  "  Empirical 
Logic  "  is  unusually  sagacious  and  suggestive  in  all  the  most 
fundamental  matters.  Jevons  receives  due  acknowledgment 
in  the  extent  to  which  I  have  modelled  my  work  after  his. 

JAMES  H.   HYSLOP. 

Columbia  College,  April  23,  1892. 


CONTENTS 

CHAPTER  I.  page 

Introduction,  Definition  and  Divisions  of  Logic,        .        .        1 

CHAPTER  II. 
Elements  of  Logical  Doctrine, 16 

CHAPTER  III. 
Terms  or  Concepts  and  their  Kinds, 31 

CHAPTER  IV. 
The  Ambiguity  of  Terms, 50 

CHAPTER  V. 
The  Intension  and  the  Extension  of  Concepts,    ...      68 

CHAPTER  VI. 
Definition  and  Division, 82 

CHAPTER  VII. 
Propositions  or  Judgments, 105 

CHAPTER  VIII. 
The  Relation  between  Subject  and  Predicate,    .        .        .     123 


x  CONTENTS 

CHAPTER  IX.  PAGE 

Opposition, .    141 

CHAPTER  X. 
Immediate  Inference, 154 

CHAPTER  XI. 
Principles  of  Mediate  Reasoning, 171 

CHAPTER  XIL 
Moods  and  Figures  of  the  Syllogism,     .  ...    181 

CHAPTER  XIII. 
Reduction  of  Moods  and  Figures, 190 

CHAPTER  XIV. 
Forms  of  Syllogistic  Reasoning, 197 

CHAPTER  XV. 
Hypothetical  Reasoning, 204 

CHAPTER]  XVI. 
Disjunctive  Syllogisms, 212 

CHAPTER  XVH. 
Classification  of  Fallacies, 219 

CHAPTER   XVIII. 
Matruial  Fallacies, 228 


CONTENTS  xi 

CHAPTER   XIX.  page 

Quantification  of  the  Predicate, 262 

CHAPTER  XX. 

Mathematical  and  other  Reasoning, 275 

CHAPTER  XXI. 

The  Laws  of  Thought, 290 

CHAPTER  XXII. 

Inductive  Reasoning 295 

CHAPTER  XXIII. 

Scientific  Method, 336 

Practical  Questions  and  Problems, 367 

Practical  Exercises, 383 

INDEX, 399 


EKRATA. 

Page  20. — In  line  7  from  the  bottom  read  belongs  for  "belong,"  and  in 

line  6  read  represents  for  "  represent." 
Page  120. — In  line  17  from  the  top  add  some  of  before  the  last  word  and 
also  after  the  word  "applying"  in  line  21.     Omit  the  sentences  in 
lines  22  to  26. 
Page  142. — In  line  2  from  the  bottom  read  contraries  for  "  opposites." 
Page  170. — In  line  10  from  the  top  read  Allnot-men  for  "No  not-men." 
Page  224. — In  illustration  for  the  illicit  process  of  the  major  term  read 

Fig.  I  for  "  Fig.  II." 
Page  354. — In  line  4  from  the  bottom  read  is  for  "  are." 
Page  374. — In  line  4  from  the  top  read  woman  for  "  women." 
Page  395. — In  the  second  example  under  deductive  and  inductive  illus- 
trations read  slowly  for  "slow." 


ELEMENTS   OF  LOGIC 


CHAPTER  I. 

INTRODUCTION— DEFINITION   AND   DIVISIONS   OF   LOGIC 

i".  DEFINITION. — Logic  lias  sometimes  been  defined  as  a 
science,  sometimes  as  an  art,  and  sometimes  as  both  a  science 
and  an  art.  Dr.  "Watts  calls  it  "  the  Art  of  Thinking ; "  Thomp- 
son, "  the  Science  of  the  Laws  of  Thought ; "  and  Whately, 
'•  the  Science  and  also  the  Art  of  Reasoning."  More  elaborate 
and  technical  definitions  are  such  as  Hamilton's,  that  "  Logic 
is  the  Science  of  the  Formal  and  Necessary  Laws  of  Thought 
as  Thought ; "  or  Ueberweg's,  that  it  is  "  the  Science  of  the 
Regulative  Laws  of  Human  Knowledge."  Most  writers  define 
it  as  a  science  instead  of  an  art,  and  in  so  far  as  they  so  regard 
it  their  views  of  it  are  substantially  the  same.  They  begin  to 
diverge  from  each  other  only  when  they  speak  of  it  as  an  art. 
Yet  a  careful  examination  of  the  various  usages  of  the  term 
"art"  will  show  that  the  difference  between  it  and  a  science 
is  less  than  general  disputes  would  imply.  This  is  apparent  if 
we  but  reflect  that  science  and  art  usually  have  the  same  sub- 
ject-matter, although  they  have  different  ends  in  view.  A 
science  teaches  us  to  know ;  an  art,  to  do.  Science  endeavors 
to  discover  truth  or  knowledge,  art  to  apply  it,  or  to  formu- 
late rules  for  applying  it,  to  the  realization  of  some  other  end. 
But  it  is  evident  in  such  a  view  that  art  assumes  knowledge  as 
a  condition  of  itself,  and  hence  science  and  art  may  go  together 
as  complementary  of  each  other.  This  is  true  of  Logic  to  the 
extent  that  it  may  be  regarded  both  as  a  science  and  as  an  art, 
according  as  it  aims  at  certain  truths,  or  at  the  application  of 


2  ELEMENTS  OF  LOGIC 

them  in  practice.  Logic  as  a  science  aims  to  ascertain  what 
are  the  laws  of  thought  ;  as  an  art  it  aims  to  apply  these  laws 
to  the  detection  of  fallacies  or  for  the  determination  of  correct 
reasoning.  As  a  science  Logic  will  be  concerned  chiefly,  if  not 
wholly,  with  the  general  principles  of  thinking,  and  only  indi- 
rectly with  truths  of  any  other  kind.  The  laws  of  thought,  the 
object  to  which  they  can  be  applied,  or  the  kind  of  phenomena 
in  which  they  are  embodied,  are  all  that  Logic  as  a  science 
need  occupy  itself  with.  Truths  or  knowledge  in  other  fields 
of  mental  interest  may  be  useful  for  illustration,  but  they  are 
not  to  be  investigated  by  it.  Only  the  truth,  extent,  nature, 
and  validity  of  the  laws  of  rational  thinking  come  under  its 
scientific  aspect.  But  when  we  wish  to  know  whether  other 
bodies  of  knowledge,  in  which  reasoning  is  involved,  have  been 
correctly  obtained  or  proved,  we  subject  them  to  the  test  of 
logical  laws,  and  to  do  this  we  may  be  obliged  to  adopt  a  large 
system  of  rules  for  practice.  The  formulation  of  such  rules 
and  the  testing  of  the  various  material  truths  of  other  sciences 
are  left  to  Logic  as  an  art.  The  truth  and  validity  of  its  gen- 
eral laws  will  be  assumed  and  admitted.  The  problem  will  be 
to  find  whether  individual  processes  of  reasoning  have  been 
conducted  in  conformity  with  those  laws  or  not.  Hence,  as  an 
art,  it  is  concerned  with  a  larger  system  of  truths  than  it  is  as 
a  science.  In  both  it  is  concerned  with  the  laws  of  thought  ; 
but  as  a  science  it  treats  those  laws  as  an  end  for  its  own  sake, 
and  as  an  art  it  treats  them  as  a  means  to  a  remoter  end. 
Hence  there  is  no  necessity  of  deciding  whether  it  is  one  or 
the  other.  The  old  controversy  concerning  that  question  may 
thus  be  settled  by  defining  the  subject  as  both  a  science  and 
an  art  under  limitations.*     Other  aspects  of  the  definition 

*  For  a  discussion  of  the  nature  of  Logic,  and  of  the  relation  between 
science  and  art,  see  Whately  :  Elements  of  Logic,  Introduction;  Fowler  : 
Elemenl  of  Deductive  Logic, Introduction, Chapter II.  ;  Thompson:  Laws 
of  Thought,  Introduction,  Section  6  ;  Hamilton:  Lectures  on  Metaphys- 
ics, Lecture  VII.-,  Lectures  in  Logic,  Lecture  I.;  .T.  S.  Mill:  Logic,  In- 
troduction ;  and  Examination  of  the  Philosophy  of  Sir  William  Hamilton, 
Chapter  XX. 


DEFINITION  AND  DIVISIONS  OF  LOGIC  3 

will  require  separate  consideration.  But  to  indicate  what 
these  are  we  shall  adopt  as  the  most  complete  definition  for 
our  purposes  that  which  makes  Logic  "the  Science  of  the 
Formal  Laws  of  Thought."  Three  terms  of  this  definition  re- 
quire special  examination. 

1st.  Thought. — "Thought,"  in  common  usage,  is  a  very 
comprehensive  term.  It  is  even  coextensive  with  consciousness 
or  mind.  We  use  the  expression,  "  I  have  such  a  thing  in  my 
thoughts,"  whereby  we  mean  merely  that  attention  perhaps  is 
occupied  with  a  particular  idea.  It  is  not  in  any  such  sense 
that  Logic  must  use  the  term.  To  have  an  idea  in  conscious- 
ness does  not  necessarily  imply  that  any  logical  processes  are 
going  on,  or  that  the  mind  is  "  thinking  "  logically  about  the 
fact.  It  may  denote  no  more  than  an  act  of  perception  or  at- 
tention. In  the  loose  sense  of  the  term,  therefore,  "  thought " 
might  denote  any  conscious  act  of  the  mind.  But  to  regard 
Logic  as  the  science  of  such  activities  and  their  laws  would 
identify  it  with  Psychology.  Hence  "thought"  must  denote 
a  more  specific  act  of  the  mind,  and  this  is  supposed  to  be  the 
act  which  compares  and  reasons.  The  term  may  denote  both 
the  act  and  the  product  of  the  rational  faculty. 

The  various  acts  of  the  mind  may  be  denominated  sensation, 
perception,  apprehension  or  cognition,  memory,  association, 
attention,  etc.  But  all  of  these  are  comparatively  simple  acts. 
They  do  not  require  any  act  of  comparison  by  the  mind. 
"  Thought,"  in  the  logical  sense,  does  require  such  compari- 
son. The  simple  perceptive  acts  of  the  mind  have  but  one 
thing  as  the  object  of  consciousness,  and  hence  denote  either 
presentations  or  individual  states  of  mind  without  taking  into 
account  any  relations  that  might  be  connected  with  them. 
"  Thought,"  on  the  other  hand,  does  explicitly  express  the 
consciousness  of  some  relation  between  two  or  more  objects 
held  together  in  consciousness  at  the  same  time.  Thus  I  may 
perceive  a  tree,  a  house,  a  man,  without  performing  any  men- 
tal act  which  thinks  them  in  relation  to  other  objects  of  a  like 
or  different  kind,  or  without  apprehending  their  meaning. 
But  if  I  think  of  a  tree  as  a  vegetable,  of  a  house  as  a  useful 


4  ELEMENTS  OF  LOGIC 

structure,  or  of  a  man  as  rational,  I  am  apprehending  the 
meaning  or  relation  of  the  several  objects,  holding  two  con- 
ceptions in  the  mind  at  the  same  time  and  pronouncing  upon 
their  connection  or  disconnection,  as  the  case  may  be.  Thus 
"thought,"  as  the  subject-matter  of  logical  science  is  an  act 
connecting  two  distinct  ideas,  or  is  the  product  of  such  an  act. 
Verbally  it  is  the  act,  nominally  it  is  the  product.  Logic  does 
not  need  specially  to  distinguish  between  them  for  its  pur- 
poses. But  it  does  require  to  consider  "  thought  "  in  its  nar- 
rower signification  as  an  act  of  comparison  between  concep- 
tions in  order  to  distinguish  more  clearly  its  own  laws  from 
the  laws  with  which  Psychology  is  concerned. 

"  The  term  thought"  says  Sir  William  Hamilton,  "  is  used 
in  two  significations  of  different  extent.  In  the  wider  mean- 
ing, it  denotes  every  cognitive  act  whatever  ;  by  some  philos- 
ophers, as  Descartes  and  his  disciples,  it  is  used  for  every 
mental  modification  of  which  we  are  conscious,  and  thus  in- 
cludes the  Feelings,  Volitions,  and  the  Desires.  In  the  more 
limited  meaning,  it  denotes  only  the  acts  of  the  Understand- 
ing properly  so  called,  that  is,  of  the  Faculty  of  Comparison, 
or  that  which  is  distinguished  as  the  Elaborative  or  Discur- 
sive Faculty.  It  is  in  this  more  restricted  signification  that 
thought  is  said  to  be  the  object-matter  of  Logic.  Thus  Logic 
does  not  consider  the  laws  which  regulate  the  other  powers 
of  mind.  It  takes  no  immediate  account  of  the  faculties  by 
which  we  acquire  the  rude  materials  of  knowledge  ;  it  supposes 
these  materials  in  possession,  and  considers  only  the  manner 
of  their  elaboration.  It  takes  no  account,  at  least  in  the  de- 
partment of  Pure  Logic,  of  Memory  and  Imagination,  or  of 
the  blind  laws  of  Association,  but  confines  its  attention  to  con- 
nections regulated  by  the  laws  of  intelligence.  Finally,  it 
does  not  consider  the  laws  themselves  of  Intelligence  as  given 
in  the  Regulative  Faculty ;"  namely,  the  Intuitions  of  pure  in- 
telligence, or  the  ultimate  data,  facts,  and  principles  which  are 
involved  in  the  primary  experiences  of  mind.  Put  such  are 
the  functions  with  which  Logic  is  not  conversant.  It  remains 
to  determine  positively  what  the  nature  of  its  object-matter  is. 


DEFINITION  AND  DIVISIONS  OF  LOGIC  5 

"  The  contemplation  of  the  world  presents  to  our  subsidiary 
faculties  a  multitude  of  objects.  These  objects  are  the  rude 
materials  submitted  to  elaboration  by  a  higher  and  self-active 
faculty,  which  operates  upon  them  in  obedience  to  certain 
laws,  and  in  conformity  to  certain  ends.  The  operation  of 
this  faculty  is  Thought.  All  thought  is  a  comparison,  a  recog- 
nition of  similarity  or  difference ;  a  conjunction  or  disjunc- 
tion ;  in  other  words,  a  synthesis  or  analysis  of  its  objects. 
In  Conception,  that  is,  in  the  formation  of  concepts  (or  gen- 
eral notions),  it  compares,  disjoins,  or  conjoins  attributes ;  in 
an  act  of  Judgment,  it  compares,  disjoins,  or  conjoins  con- 
cepts; in  Reasoning,  it  compares,  disjoins,  or  conjoins  judg- 
ments. In  each  step  of  this  process  there  is  one  essential 
element ;  to  think,  to  compare,  to  conjoin,  or  disjoin,  it  is  nec- 
essary to  recognize  one  thing  through  or  under  another  ;  and 
therefore  in  defining  Thought  proper,  we  may  either  define  it 
as  an  act  of  Comparison,  or  as  a  recognition  of  one  notion  as 
in  or  under  another.  It  is  in  performing  this  act  of  thinking 
a  thing  under  a  general  notion,  that  we  are  said  to  understand 
or  comprehend  it.  For  example,  an  object  is  presented,  say  a 
book  ;  this  object  determines  an  impression,  and  I  am  even 
conscious  of  the  impression,  but  without  recognizing  to  myself 
what  the  thing  is  ;  in  that  case  there  is  only  perception,  and 
not  properly  a  thought.  But  suppose  I  do  recognize  it  for 
what  it  is,  in  other  words,  compare  it  with,  and  reduce  it 
under,  a  certain  concept,  class,  or  complement  of  attributes, 
which  I  call  book ;  in  that  case,  there  is  more  than  a  percep- 
tion— there  is  a  thought." 

Thought  is,  therefore,  the  act  or  product  of  the  Under- 
standing or  Reason  as  distinct  from  the  various  processes  of 
simple  Apprehension  or  Cognition,  and  consequently  Logic  is 
conversant  with  the  laws  affecting  or  regulating  this  act  of 
comparative  knowledge  rather  than  with  the  laws  of  percep- 
tion. It  is  the  science  of  thought  as  an  act  of  conception, 
judgment,  and  reasoning,  or  as  the  cognition  of  relations  be- 
tween conceptions.  This  act  has  its  own  laws  distinct  from 
those  of  other  mental  acts.     But  the  limitations  of  the  science 


6  ELEMENTS  OF  LOGIC 

and  its  laws  can  be  determined  only  by  comparing  its  field 
and  functions  with  those  of  other  sciences.  The  nature  of  its 
general  object-matter  suffices  to  define  and  distinguish  it  from 
cognate  sciences,  on  the  one  hand,  and  from  the  physical  sci- 
ences on  the  other.  "  Thought "  is,  first,  a  fact  quite  distinct 
from  physical  events,  and  is,  second,  in  its  technical  sense,  as 
logically  defined,  distinct  from  mental  events  which  are  not 
acts  of  comparison  :  hence  it  implies  a  double  limitation  of  the 
subject  of  Logic  ;  one  its  distinction  from  sciences  which  in- 
vestigate the  physical  laws  and  causes  of  events,  and  the  other 
its  distinction  from  the  philosophical  sciences  which  are  occu- 
pied either  with  the  efficient  causes  of  mental  phenomena  or 
with  the  nature  of  the  being  of  which  they  are  phenomena. 
Logic  is  occupied  with  the  relations  between  those  phenomena, 
as  objects  of  reason  or  rational  thought,  or  with  the  laws 
wThich  attest  the  validity  of  thought,  and  which  serve,  at  least, 
as  negative  criteria  of  the  truth  in  so  far  as  it  is  determined 
by  processes  of  reasoning.  This,  however,  will  be  rendered 
clearer  when  we  examine  the  relation  of  Logic  to  the  various 
sciences. 

2d.  The  Nature  of  the  Laws  of  Thought. — The  laws  of 
thought  will  be  best  understood  by  defining  the  several  uses  of 
the  term  "law."  That  there  are  "laws"  of  thought  we  may 
at  present  take  for  granted,  unless  we  are  to  assume  that 
there  are  no  regulative  principles  or  conditions  which  deter- 
mine the  uniformity  of  the  mental  processes  involved  in  the 
various  acts  of  thought.  But  the  assumption  that  there  are 
"laws"  of  thought  neither  defines  their  specific  nature  nor 
affords  any  indication  of  what  they  are.  This  desirable  end 
can  be  achieved  only  by  an  examination  of  what  we  mean  by 
"  law." 

The  first  of  the  two  general  meanings  of  the  term  is  of  little 
importance  to  a  discussion  in  Logic,  except  that  it  may  be' 
serviceable  for  bringing  the  true  meaning  of  it  into  proper 
relief.  But  this  first  conception  is  its  moral  and  political 
sense,  in  which  it  denotes  a  command  or  prohibition  in  regard 
to  the  doing  of  certain  actions.     This  idea  does  not  imply  any 


DEFINITION  AND   DIVISIONS  OF  LOGIC  7 

absolute  necessity  of  the  event  commanded,  nor  does  it  imply 
any  regularity  of  such  events.  It  denotes  only  an  injunction 
to  act  or  not  to  act.  But  such  a  conception  is  quite  the  con- 
trary of  the  idea  of  "  law  "  as  employed  by  the  sciences,  where 
it  denotes  the  regularity  with  which  events  occur,  or  the  uni- 
formity of  their  dependence  upon  certain  conditions  which  ne- 
cessitate them.  Hence  the  notion  of  "  law,"  as  applied  in  the 
physical  sciences,  is  but  an  abbreviation  for  the  uniformity  of 
coexistence  and  sequence,  or  the  uniformity  of  causation.  It 
describes  or  implies  those  conditions  which  make  events  regu- 
lar and  inevitable,  or  which  explain  why  they  follow  a  given 
order  of  occurrence.  It  is  thus  opposed  to  chance  or  caprice. 
In  so  far  as  the  idea  is  synonymous  with  the  notion  of  cause, 
it  is  a  convenient  term  for  referring  to  the  explanation  of 
phenomena.  In  so  far  as  it  denotes  merely  uniformity  of 
events,  it  is  convenient  for  indicating  the  general  principles 
of  unity  that  are  exhibited  or  expressed  thereby  in  the  multi- 
plicity of  phenomenal  events. 

But  this  form  of  discussion  is  perhaps  not  quite  so  clear  as 
illustration.  Each  of  the  sciences  endeavors  to  ascertain  the 
laws  which  prevail  in  a  special  class  of  phenomena.  Chem- 
istry tries  to  find  out  the  laws  of  affinity  and  combination 
regulating  the  relations  between  atomic  bodies.  Thus  oxy- 
gen and  hydrogen  must  combine  in  a  certain  proportion  to 
produce  water,  and  they  will  combine  in  no  other  proportion 
to  produce  the  same  effect.  This  represents  a  certain  "  law  " 
of  affinity  between  the  elements.  A  similar  law  operates 
among  all  elements  which  combine  only  in  certain  definite 
proportions  which  can  be  expressed  with  perfect  mathematical 
accuracy.  In  Astronomy  we  speak  of  the  law  of  gravitation 
which  expresses  the  uniform  tendency  of  material  bodies  to 
seek  the  centre  of  gravity,  or  to  move  toward  it,  when  free, 
with  a  determinate  velocity.  In  Physics  we  have  the  laws 
of  the  expansion  and  diffusion  of  gases  ;  of  contraction  and 
expansion  under  changes  of  temperature  ;  of  the  conservation 
of  energy,  and  of  the  transmission  of  motion.  In  all  these 
cases  there  are  certain  fixed  ways  of  acting  which  bodies  must 


8  ELEMENTS  OF  LOGIC 

follow  or  obey.  All  facts  must  be  regulated  or  determined  by 
the  conditions  which  such  laws  impose. 

But  there  is  a  science  of  human  reason,  and  the  processes 
of  thought  are  as  much  under  law  as  any  other  phenomena. 
These  laws,  like  all  other  laws  or  conditions  of  events,  are  uni- 
formities of  nature  ;  in  this  case,  of  the  nature  of  the  mind. 
They  are  embodied  in  principles  or  propositions  which  show 
how  we  must  think  and  reason,  if  we  think  and  reason  at  all. 
They  are  necessary  laws  of  thought,  because  the  mind  has  no 
power  to  evade  them.  Thus  conception,  judgment,  and  rea- 
soning are  according  to  a  principle  which  conditions  and  vali- 
dates them,  and  so  is  known  as  their  law,  or  the  uniformity 
of  mental  action  in  them  which  enables  us  to  accept  their 
results.  Such  laws  may  be  illustrated  in  the  following  man- 
ner : 

If  I  take  the  judgment  that  "  All  men  are  mortal,"  and  infer 
from  it,  by  a  process  which  is  yet  to  be  considered,  that 
"  those  who  are  not  mortal  are  not  men,"  or,  "  The  immortals 
are  not  men,"  and  if  I  can  apply  a  similar  process  to  all  such 
propositions,  it  is  because  of  the  law  that  what  is  excluded  from 
the  conception  "  mortal",  must  be  excluded  from  the  conception 
"  men."  The  assumption  of  this  law  is  necessary  to  the  mak- 
ing of  this  inference,  and  unless  it  always  be  true  I  have  no 
means  of  guaranteeing  the  legitimacy  of  the  results.  Again,  I 
may  take  three  conceptions  which  are  capable  of  agreeing 
with  each  other  when  conjoined  in  the  form  of  a  judgment. 

Metals.  Iron.  Useful. 

By  connecting  these  as  subject  and  predicate  I  obtain  three 
propositions,  one  of  which  is  a  conclusion  or  inference  from 
the  other  two.     Thus  : 

Metals  are  useful.  Iron  is  a  metal.  .  ■ .  Iron  is  useful. 
Illustrations  could  be  multiplied  indefinitely  showing  the 
same  form  and  process  of  thinking,  but  the  one  suffices  to  call 
attention  to  the  principle  upon  winch  the  mental  act  is  based. 
We  observe  that  in  the  first  two  propositions  there  is  a  com- 
mon term  compared  with  two  others.  The  first  proposition 
states  an  agreement  between  "  metals"  and  "  useful,"  and  the 


DEFINITION  AND  DIVISIONS  OF  LOGIC  9 

second  between  "  iron "  and  "  metals."  The  common  term 
with  which  "iron"  and  "useful"  agree  is  " metals,"  and  this 
fact  affords  a  reason  to  suppose  that  "  iron  "  and  "  useful " 
also  agree  to  the  same  extent  and  in  the  same  sense.  If  we 
find  that  all  such  cases  of  reasoning  exemplify  the  same  form 
of  process  and  conditions  we  may  formulate  the  law  which 
regulates  and  legitimates  it.  It  is  a  logical  axiom.  Thus,  if 
two  things  are  identical  or  agree  with  a  third  common  thing,  they 
are  identical  or  agree  with  each  other.  This  is  a  fundamental 
law  of  thought  and  sustains  the  same  relation  to  the  validity 
of  reasoning  that  the  law  of  gravitation  sustains  to  the  phe- 
nomena of  falling  bodies  or  the  motion  of  the  planets  about 
the  sun.  It  is  a  law,  because  it  expresses  the  uniform  way  in 
which  the  mind  acts,  and  must  act,  when  comparing  concep- 
tions and  judgments.  There  are  many  other  such  laws,  but 
they  do  not  require  examination  at  this  stage  of  the  discus- 
sion. One  illustration  suffices  to  show  what  it  is  that  Logic 
endeavors  to  investigate  and  establish.  Its  laws  are  generally 
expressed  as  principles  or  assumptions,  while  those  of  the 
physical  sciences  are  usually  conceived  as  causes,  conditions, 
or  uniformities  of  coexistence  and  sequence.  But  in  both,  the 
essential  idea  is  uniformity  of  some  kind.  In  the  physical 
sciences  it  is  the  regularity  with  which  certain  events  occur, 
and  may  be  expected  to  occur,  under  given  conditions.  In 
logical  science,  the  uniformity  is  that  of  mental  action  when 
thinking  and  reasoning,  as  the  fixed  modes  of  comparison  and 
inference  which  the  mind  must  obey  when  its  action  is  healthy 
and  rational.  In  that  respect  Logic  is  a  science  like  all  other 
forms  of  inquiry  into  the  nature  and  principles  of  phenomena  ; 
only  its  laws  are  the  laws  of  reason,  and  not  of  physical  events. 
3d.  Form  and  Matter. — The  distinction  between  "form" 
and  "  matter,"  and  the  definition  of  the  term  "  formal "  as 
employed  in  Logic,  are  two  of  the  most  important  things  to 
be  accomplished,  as  their  peculiar  signification  appears  to 
influence  many  of  the  doctrines  of  the  science  and  to  explain 
many  of  the  perplexities  which  are  incident  to  logical  pro- 
cesses.    The  term  "  formal  "  we  shall  find  often  to  be  equiva- 


10  ELEMENTS  OF  LOGIC 

lent  in  many  respects  to  the  idea  of  "law."  The  "  forms  "  of 
thought  are  often  called  the  laws  of  thought,  and  they  are 
such  in  their  nature.  But  why  they  are  so  can  be  more  dis- 
tinctly understood  when  we  have  made  clear  the  difference  be- 
tween the  "form  "  and  the  "matter"  of  thought,  or  between 
the  uses  of  those  terms  in  reference  to  any  subject-matter 
whatever. 

What  we  mean  by  "  form "  when  applied  to  material  or 
physical  objects  is  generally  clear  enough.  It  is  synonymous 
with  shape,  the  geometrical  outline  of  a  body,  or  the  special 
limitations  under  which  a  physical  object  is  conceived.  The 
"  form  "  of  a  body  may  be  regular  or  irregular,  long  or  short, 
round,  square,  rectangular  thick,  thin,  flat,  etc.  But  no  such 
predicates  can  be  apjilied  to  thought,  or  to  states  of  con- 
sciousness. Mental  states  can  neither  be  said  to  have  "form  " 
in  the  sense  of  having  the  quality  of  extension,  nor  to  exist  in 
any  assignable  relation  to  space.  Hence  we  cannot  speak  of 
their  "  form "  as  we  would  of  material  objects.  But  inas- 
much as  the  "form"  of  physical  objects  does  not  necessarily 
depend  upon  the  stuff  or  matter  of  which  they  consist,  but 
may  remain  the  same  amid  all  changes  of  the  latter,  or  the 
same  when  the  matter  may  be  of  different  kinds,  because  ex- 
tension is  independent  of  material  substances,  this  conception 
of  the  case  may  be  chosen  to  describe,  at  least  by  analogy, 
certain  relations  between  the  processes  and  the  objects  of 
thought  or  consciousness.  The  fact  first  suggests,  however, 
the  definition  of  "matter,"  which  in  its  relation  to  "form,"  as 
applied  to  physical  objects,  is  merely  the  physical  elements, 
apart  from  extension,  which  make  a  body  or  object  what  it  is 
other  than  its  "  form."  It  is,  as  already  said,  the  stuff  of  which 
it  consists.  But  by  the  same  analogy  which  transfers  the  use 
of  the  term  "  form  "  to  other  than  physical  objects  the  term 
"  matter  "  is  also  transferred,  and  the  two  terms  are  chosen  to 
express  certain  relations  of  uniformity  and  variation  existing 
between  all  kinds  of  phenomena,  physical  or  mental.  Hence, 
as  two  things  may  be  alike  in  their  "  form,"  while  they  are 
of  different  "  matter  "  or  substance,  two  mental  processes  may 


DEFINITION  AND  DIVISIONS  OF  LOGIC  11 

be  the  same  in  kind,  although  occupied  about  different  ob- 
jects of  thought.  Thus  the  judgments,  "All  men  are  mortal," 
and  "  All  metals  are  elements,"  are  alike  in  the  one  aspect 
of  grammatical  structure,  but  are  different  in  respect  to  the 
conceptions  of  which  they  are  composed.  Two  or  more  syllo- 
gisms consisting  of  different  propositions  will  illustrate  the 
same  truth.  The  oue  aspect  in  which  they  resemble  each 
other  may  be  called  their  "  form,"  and  the  conceptions  which 
compose  them,  and  which  may  vary  indefinitely  without  affect- 
ing the  "  form,"  grammatical  or  logical,  may  be  called  their 
"matter."  Hence  we  may  define  the  "form,"  at  least  provi- 
sionally,  that  which  remains  constant  when  the  "matter"  changes, 
and,  correlative!}-,  the  "matter"  may  be  defined  as  that  which 
remains  constant  when  the  "form"  changes. 

Objections  can  be  made  to  these  definitions,  the  force  of 
which  can  be  readily  admitted.  But  they  are  intended  to  be 
merely  tentative  and  approximate,  and  for  the  purpose  of 
making  them  general  enough  to  cover  all  conceivable  objects 
of  consciousness.  It  is  a  misfortune  to  be  compelled  to  define 
the  terms  reciprocally.  But  their  simplicity  and  the  fact  that 
they  are  purely  relative  to  each  other  is  an  excuse  for  such  a 
course,  and  may  even  make  it  necessary.  All  objects,  real  or 
ideal,  must  have  both  their  "matter"  and  their  "form"  at 
the  same  time.  But  the  fact  that  either  quality  may  be  con- 
stant while  the  other  is  variable  proves  the  value  of  defining 
them  in  a  way  to  recognize  both  this  variable  relation  and  the 
permanent  coexistence  of  the  two  qualities  in  one  way  or  an- 
other at  the  same  time. 

But  the  "  form"  is  usually  regarded  as  what  is  essential  to  a 
thing  as  well  as  what  is  constant.  This  means  that  a  certain 
quality  cannot  be  dispensed  with  in  reference  to  a  given  end, 
although  others  may  be  immaterial  or  accidental  in  this  re- 
spect. Thus  a  house  may  serve  the  same  purpose  whether 
made  of  brick  or  wood,  while  the  "form  "  may  not  only  be  the 
same,  but  must  be  more  or  less  essential.  A  judgment  must 
have  a  subject  and  predicate,  whatever  the  conceptions  con- 
stituting it,  and  so  the  "form"  is  essential  to  its  being  a  judg- 


12  ELEMENTS  OF  LOGIC 

merit,  while  its  matter  is  not.  This  idea  of  what  is  essential 
must  be  added  to  that  of  constancy  in  order  to  complete  the 
notion  of  "  form  "  and  its  distinction  from  "  matter." 

But  it  is  well  to  remark  again  the  purely  relative  character 
of  the  two  conceptions  in  order  to  anticipate  and  remove  a 
possible  objection.  The  two  qualities  are  so  variable  that 
what  is  the  "  form  "  in  one  relation  may  become  the  "  matter" 
in  another.  It  depends  wholly  upon  the  nature  of  the  relation 
in  which  an  object  is  conceived.  The  fact,  however,  does  not 
require  development  in  an  elementary  treatise  of  the  science. 
It  is  noticed  only  to  indicate  that  the  circumstance  has  not 
been  overlooked  in  our  account  of  the  subject  under  consid- 
eration. 

The  relation  between  "form  "  and  "matter,"  as  we  have  de- 
fined it,  expresses  very  clearly  how  Logic  is  a  science  of  the  for- 
mal laws  of  thought.  They  are  the  laws  which  are  not  only 
essential  to  it,  but  which  are  the  same  whatever  the  subject- 
matter  involved  in  our  reasoning.  The  laws  of  thought  remain 
the  same  in  the  reasonings  of  Astronomy,  Physics,  Politics,  or 
Ethics,  but  the  "  matter  "  changes  and  does  not  affect  the  va- 
lidity of  the  process.  The  "  form  "  of  our  reasoning  in  all 
these  cases  is  essential  to  its  being  such  a  process.  Hence 
Logic,  as  a  science,  is  "formal,"  and  deals  only  with  the  "for- 
mal "  principles  of  thought  in  distinction  from  the  material 
objects  of  reason.  Logic  thus  becomes  the  most  general  of  all 
the  sciences,  and  its  principles  are  the  "  formal "  conditions  of 
the  truth  attained  in  them.  What  these  "forms"  of  thought 
are  will  appear  with  the  development  of  the  science.  It  is  only 
important  at  present  to  know  that  we  do  not  require  to  take 
any  account  of  the  particular  "  matter"  of  knowledge  in  order 
to  ascertain  these  "formal"  laws. 

//.  TEE  RELATION  OF  LOGIC  TO  THE  SEVERAL  SCI- 
ENCE8. — Logic  we  have  found  to  be  the  science  of  the  formal 
laws  of  thought.  This  fact  has  led  to  its  definition  as  the  Sci- 
ence of  Sciences  (scientia  scientiarum).  This  is  true  if  consid- 
ered only  in  its  formal  functions.  Any  other  conception  of  it 
would  imply  that  it  was  concerned  with  their  material  truths. 


DEFINITION  AND  DIVISIONS  OF  LOGIC  13 

But  it  can  employ  itself  with  the  material  knowledge  of  other 
sciences  only  in  its  function  as  an  Art  (ars  artium).  As  a  sci- 
ence, however,  it  has  a  "  formal "  relation  to  all  other  sciences 
inasmuch  as  it  determines  the  laws  of  thought  everywhere 
affecting  the  acquisition  and  legitimation  of  truth.  This  is 
all  that  it  is  necessary  to  recognize  in  a  general  way.  But  in- 
asmuch as  it  sustains  a  more  intimate  and  complex  relation  to 
the  mental  sciences,  often  dealing  with  precisely  the  same  sub- 
ject-matter, we  require  to  distinguish  between  its  relation,  in 
some  particulars,  to  the  physical  sciences,  and  its  relation  to 
the  mental.  While  it  formally  conditions  the  mental  processes 
of  the  physical  sciences,  it  is  materially  occupied  with  the 
laws  of  thought,  and  they,  the  physical  sciences,  with  the  laws 
of  things.  The  other  mental  sciences,  however,  to  which  it 
has  a  more  peculiar  relation  beside  the  general  formal  one, 
have  quite  a  distinct  object  in  view  as  compared  with  Logic. 
The  two  sciences  are  Psychology  and  Ethics.  With  Logic 
these  constitute,  properly,  the  mental  sciences. 

Psychology  deals  with  all  mental  phenomena  ;  Logic  deals 
with  only  a  part  of  them.  They  therefore  partly  coincide.  In 
so  far  as  they  both  deal  with  the  thought  processes  of  judg- 
ment and  reasoning  they  are  occupied  with  the  same  field. 
But  they  do  not  deal  with  these  phenomena  in  the  same  way, 
nor  with  the  same  object  in  view.  "  Psychology  deals  with 
them  as  laws  in  the  sense  of  uniformities,  that  is,  as  laws  in  ac- 
cordance with  which  men  are  found  by  experience  normally  to 
think  and  reason  ;  Psychology  investigates  also  their  genesis 
and  origin.  Logic,  on  the  other  hand,  deals  with  them  purely 
as  regulative  and  authoritative,  as  affording  criteria  by  the  aid 
of  which  valid  and  invalid  reasonings  may  be  discriminated, 
and  as  determining  the  formal  relations  in  which  the  different 
products  of  thought  stand  to  one  another."  These  observa- 
tions of  Keynes  may  perhaps  be  rendered  a  little  more  com- 
plete if  we  remark  that  Psychology  investigates  all  mental 
phenomena,  including  the  rational,  with  a  view  to  ascertain ing, 
first,  their  uniformities  as  actual  events,  and  second,  their 
causes,  but  does  not  require  to  distinguish  between  normal  and 


14  ELEMENTS  OF  LOGIC 

abnormal  states,  valid  or  invalid  ideas.  On  the  other  hand, 
Logic,  in  addition  to  being  conversant  with  a  more  limited 
field  than  Psychology,  does  not  deal  with  the  phenomena  in 
common  to  the  two  sciences  with  the  same  object  in  view 
throughout.  It  deals  with  the  uniformities  of  rational  pro- 
cesses, but  neither  with  their  causes,  nor  with  any  matters  re- 
garding the  origin  of  those  uniformities.  It  endeavors  to  as- 
certain the  uniformities  of  thought  which  are  the  grounds  of 
truth  or  valid  thought,  and  which  exclude  all  other  ideas  from 
recognition,  except  as  facts.  Psychology  is  thus  concerned 
with  the  origin  and  nature  of  mental  phenomena  in  general, 
and  Logic  with  the  conditions  and  the  validity  of  the  rational 
processes.  The  latter,  therefore,  has  to  do  with  the  formal  in 
contradistinction  to  the  efficient  principles  of  thought.  Ethics, 
of  course,  is  the  science  of  the  ends  of  conduct,  and  although 
it  is  formally  conditioned  by  Logic,  the  relation  between  it 
and  Logic  is  not  so  close  as  between  Logic  and  Psychology. 
But  their  respective  spheres  can  be  very  well  defined  by  the 
Aristotelian  and  scholastic  formulas  expressing  the  various 
kinds  of  causes.  This  will  characterize  the  difference  between 
them  as  sciences.  Consequently  Psychology  is  the  science  of 
the  efficient  causes  (causa  efficiens,  ratio  fiendi)  of  mental  phe- 
nomena, including  the  rational  processes :  Logic  is  the  sci- 
ence of  the  formal  causes  (causa  formalis,  ratio  essendi)  of 
thought,  of  valid  rational  processes :  Ethics  is  the  science  of 
the  final  causes  (causa  finalis,  ratio  agendi)  of  human  conduct, 
of  the  mental  phenomena  of  desire  and  volition. 

It  will  thus  be  seen  how  Logic  and  Psychology  may  to  some 
extent  overlap  each  other.  This,  however,  is  mainly  in  regard 
to  the  phenomena  with  which  they  deal,  and  not  with  regard 
to  the  objects  which  they  aim  to  accomplish.  They  both  deal 
with  uniformities  of  mental  phenomena.  But  Psychology  as 
such  does  not  distinguish  between  the  true  and  the  false  ;  it 
explains  mental  processes  ;  Logic  validates  those  of  reasoning 
and  ascertains  the  principles  which  condition  them.  Psychol- 
ogy deals  with  causes,  Logic  with  principles.  This  will  per- 
baps  show  the  intimate  relation  subsisting  between  the  two 


DEFINITION  AND  DIVISIONS  OF  LOGIC  15 

sciences,  and  at  the  same  time  indicates  the  difference  of 
method  and  object  pursued  by  them  in  their  investigations. 

///.  THE  DIVISIONS  OF  LOGIC— Logic  may  be  divided 
according  to  the  particular  object  which  it  has  in  view,  or  ac- 
cording to  the  kind  of  reasoning  with  which  it  deals.  The 
first  principle  of  division  gives  us  Pure  or  Formal  Logic,  and 
Material  or  Applied  Logic.  The  same  distinction  is  observed 
in  the  divisions,  Theoretical  and  Practical  Logic.  These  divis- 
ions are  based  upon  the  distinction  between  Logic  as  a  sci- 
ence and  Logic  as  an  art.  Pure  or  Formal  Logic  is  a  science  ; 
Material  or  Applied  Logic  is  an  art.  The  former  is  conversant 
only  with  the  pure  or  formal  laws  of  thought,  and  does  not 
concern  itself  with  the  material  truth  of  any  particular  propo- 
sition ;  the  latter  is  conversant  with  the  material  conceptions 
of  various  sciences  and  endeavors  to  apply  formal  laws  to  the 
attestation  of  truth  or  knowledge.  Pure  Logic  is  abstract  and 
theoretical ;  applied  Logic  is  concrete  and  practical. 

The  second  principle  of  division  gives  us  Deductive  and  In- 
ductive Logic.  Deductive  Logic  is  usually  defined  as  occu- 
pied with  the  laws  of  a  priori  reasoning,  and  Inductive  Logic 
with  the  laws  of  a  posteriori  reasoning.  The  one  assumes  gen- 
eral principles  or  facts  in  order  to  elicit  into  consciousness 
something  which  is  not  explicitly  there  in  the  premises,  or 
which,  when  it  is  in  consciousness,  recpiires  a  general  assump- 
tion to  validate  it  as  truth  ;  the  other  ass^m/>p  facts  and  en- 
deavors to  arrive  at  general  truths  beyond  them,  or  ideas  not 
directly  deducible  from  the  premises.  But  this  division  of  the 
subject  can  be  adjusted  to  the  former,  as  the  following  scheme 
will  illustrate  : 

!„  -c         ,  (  Deductive. 

Pure  or  Formal         <  T    ,     .. 
/  Inductive. 
.      ..    ,        „  .     .  ,  (  Deductive. 
Applied  or  Material  -J  Inductive- 


CHAPTER  n. 
ELEMENTS  OP  LOGICAL  DOCTRINE 

I.  GENERAL  PRINCIPLES.  —  The  definition  of  logic 
teaches  us  that  the  science  is  employed  about  the  laws  of 
thought.  Thought,  we  also  found  to  be  an  act  of  comparison, 
and  as  the  object  of  Logic  and  logical  doctrine  it  is  usually 
identified  with  the  processes  of  reasoning.  But  all  the  higher 
and  more  complex  mental  processes  presuppose  the  lower,  and 
the  material  which  they  furnish.  Thus  reasoning  deals  with 
ideas,  notions,  conceptions,  etc.,  which  may  be  called  its  ele: 
ments.  We  cannot  understand  the  nature  of  that  j)rocess  un- 
til we  know  the  nature  of  the  elements  involved  in  it.  It  may 
be  the  function  of  Psychology  to  tell  us  how  these  elements 
have  originated,  but  it  may  not  be  its  function  to  tell  us  what 
they  are,  or  what  their  relation  to  reasoning  is.  Logic  must 
know  what  the  qualities  of  its  elements  are,  in  order  to  formu- 
late its  laws  about  their  relations  in  the  processes  of  thought. 
Hence  we  proceed  to  inquire  what  the  elements  of  logical 
doctrine  are. 

There  are  two  distinct  ways  of  regarding  this  question,  ac- 
cording as  we  consider  the  objects  of  logical  science  or  the 
mental  processes  concerned  in  it.  Hence  we  may  divide  the 
elements  of  logical  doctrine  into  two  classes  :  (a)  The  matter 
of  logical  science  which  consists  of  Terms,  Propositions,  and 
Syllogisms  ;  and  (b)  the  form  or  process  of  thought  which  is 
the  principal  object  of  the  science,  and  which  is  usually  rep- 
resented as  consisting  of  three  subordinate  mental  modifica- 
tions ;  namely,  Conception,  Judgment,  and  Reasoning.  The 
former  looks  at  the  question  from  the  point  of  view  of  lan- 
guage, and  makes  no  special  examination,  introspectively  or 
otherwise,  into  the  mental  activities  of  which  terms,  proposi- 


ELEMENTS  OF  LOGICAL  DOCTRINE  17 

tions,  and  syllogisms  are  the  object.  It  simply  relies  upon 
the  normal  accuracy  of  mental  processes  and  develops  Logic 
as  a  practical  doctrine  concerning  the  proper  mode  of  con- 
ducting arguments  and  avoiding  error.  But  from  the  point 
of  view  of  the  process  of  thought  there  is  a  desire  to  get  a 
deeper  knowledge  of  the  way  in  which  the  mind  conceives  and 
uses  its  material,  and  of  the  actions  involved  in  understanding 
the  nature  and  relations  of  its  material  objects.  Thus  we  may 
produce  a  Logic  without  entering  into  any  analysis  of  terms, 
propositions,  etc.,  and  without  considering  the  nature  of  the 
mental  processes  involved.  On  the  other  hand,  a  complete 
conception  of  the  problem  is  not  possible  until  we  have  formed 
some  definite  idea  of  the  process  as  well  as  the  matter  of 
thought.  The  two  aspects  may  be  treated  together,  as  they 
imply  each  other.  But  their  peculiarities  may  be  best  exhib- 
ited by  viewing  them  apart. 

II.  DEFINITION  OF  THE  MATERIAL  ELEMENTS  OF 
LOGICAL  DOCTRINE.  —  1st.  Terms.— A  term  may  be  vari- 
ously defined.  Grammatically  considered  it  is  a  word  which 
is  the  name  of  an  idea,  or  conception.  Logically  considered 
it  is  any  word  or  group  of  words  which  is  capable  of  constitut- 
ing a  distinct  part  of  a  sentence  or  a  proposition.  A  gram- 
matical term  is  single;  such  as  "man,"  "tree,"  "with," 
"  from,"  "  walk,"  etc.  A  logical  term  may  be  either  a  single 
word,  or  a  phrase  ;  such  as  "man,"  "house,"  or  "the  wife  of 
Socrates,"  "the  Queen  of  England."  The  singleness  of  the 
idea  is  all  that  is  required  to  make  it  a  logical  term.  But 
terms  are  the  elements,  or  atoms,  as  it  were,  which  combine 
to  form  a  proposition.  Logic  may  deal  with  them  to  some  ex- 
tent without  considering  the  processes  implied  in  apprehend- 
ing their  meaning.  This  fact  has  given  rise  to  the  belief  in 
some  schools  of  thought  that  the  science  is  exclusively  occu- 
pied about  language.  But  very  little  observation  is  required 
to  perceive  that  it  can  deal  with  language  only  as  a  symbol  of 
thought. 

2d.  Propositions. — A  proposition,  grammatically  consid- 
ered, is  a  sentence  consisting  of  words  or  terms  arranged  to 


18  ELEMENTS  OE  LOGIC 

make  sense.  Logically  considered  it  is  a  connection  of  terms, 
as  subject  and  predicate,  in  a  way  to  express  their  agreement 
or  disagreement  with  each  other.  Thus  "Iron  is  an  element," 
and  "  The  security  of  life  is  one  of  the  primary  objects  of  gov- 
ernment," are  propositions  respectively  of  the  simpler  and  the 
more  complex  form.  They  express  a  relation  between  the  sub- 
ject and  predicate,  which  cannot  be  expressed  by  a  term. 

3d.  Syllogisms. — A  syllogism  is  purely  an  object  of  Logic. 
It  is  the  combination  of  three  or  more  propositions  in  such  a 
relation  to  each  other  that  the  last  one  is  an  inference  or  con- 
clusion from  the  others.  It  is  a  simple  syllogism  if  it  consists 
of  three  propositions,  and  a  conrplex  one  when  it  consists  of 
more  than  three.  It  is  the  material  form  which  all  our  rea- 
soning assumes,  and  various  characteristics  of  it  which  are  yet 
to  be  examined  make  it  the  principal  object  of  the  investiga- 
tions conducted  by  logical  science. 

III.  EXAMINATION  OE  THE  FORMAL  ELEMENTS  OF 
LOGICAL  DOCTRINE. — As  we  have  already  ascertained,  the 
formal  elements  are  Conception,  Judgment,  and  Reason- 
ing. The  last  two  are  more  conspicuous  objects  of  logical  sci- 
ence inasmuch  as  their  laws  are  more  readily  determined 
and  the  processes  themselves  can  be  more  easily  regulated. 
The  process  of  Conception  is  so  spontaneous  and  follows  so 
closely  the  psychological  laws  of  mental  phenomena  that  con- 
cepts are  always  found  and  completed  before  logical  science 
can  take  any  account  of  them.  There  are,  therefore,  no  well- 
defined  laws  of  Conception  which  we  can  regard  beforehand 
in  the  making  of  the  primary  elements  with  winch  Judgment 
and  Reasoning  have  to  deal.  The  qualities  of  its  products, 
namely,  concepts,  can  be  observed  and  their  influence  upon 
different  mental  processes  considered.  But  it  is  more  difficult 
to  formulate  any  laws  by  which  Conception  is  governed.  Yet 
it  has  its  own  laws,  although  they  are  not  usually  formulated 
or  discussed,  except  in  Psychology.  Logicians  are  usually 
content  with  a  statement  of  the  characteristics  of  various  no- 
tions, ideas,  or  conceptions,  and  hence  they  omit  any  detailed 
account  of  the  processes  involved  in  their  formation.     But  we 


ELEMENTS  OE  LOGICAL  DOCTRINE  19 

shall  give  the  subject  a  little  more  attention  in  the  present 
treatise.  We  proceed,  therefore,  to  a  careful  examination  of 
each  pi*ocess. 

1st.  Conception. — The  term  "  conception  "  is  an  ambigu- 
ous one.  It  is  used  sometimes  verbally,  and  sometimes  nom- 
inally or  substantively.  Hence  it  may  denote  either  a  }jrocess 
or  a  product.  At  present  we  are  concerned  with  it  only  as  a 
process,  and  its  object  or  product  will  appear  for  more  partic- 
ular consideration  in  a  moment.  But  the  product  of  concep- 
tion as  a  process  is  called  a  concept.  For  the  purposes  of 
Logic  a  concept  is  the  same  as  a  term.  In  itself  it  is  to  be 
viewed  from  the  mental  side,  and  represents  the  ideal  rather 
than  the  symbolical  element  of  thought.  ^A  word  or  term  is 
only  symbolical  of  ideas,  an  index  of  thought  ;  a  concept  ex- 
presses the  content  of  the  mental  act  as  conceived  apart  from 
language,  and  so  is,  to  some  extent  at  least,  ideal  as  opposed 
to  real.  It  expresses  more  clearly  than  "  term  "  what  is  con- 
noted or  denoted  by  a  word.  The  distinction  between  it  as  a 
product,  and  conception  as  a  process,  must  be  kept  clear  if  we 
arc  to  understand  the  unity  and  simplicity  of  the  latter  as 
compared  with  the  diversity  and  complexity  of  the  former. 
Hence  we  must  define  them  as  accurately  as  possible. 

1.  Definition  of  Conception  and  Concept. — Conception  is 
the  mental  act  of  connecting  percepts  or  individual  ideas  into  a 
whole.  This  whole  may  be  of  several  kinds,  as  we  shall  presently 
see.  But  the  act  is  illustrated  in  the  process  of  conceiving 
"man,"  for  example,  as  uniting  the  properties  of  animal it;/  and 
rationality.  These  terms  may  denote  a  whole  group  of  quali- 
ties. For  instance,  "  animality  "  may  denote  material  proper- 
ties, such  as  solidity,  color,  form,  etc.,  organic  properties,  such 
as  peculiar  physiological  structure,  assimilative,  circulator}', 
respiratory  organs,  etc.,  and  "  rationality  "  may  denote  sensi- 
bility, consciousness,  reasoning  capacity,  etc.  All  these  prop- 
erties are  grouped  together  in  the  notion  of  man.  The  act 
of  mind  which  conceives  them,  or  thinks  them  as  combined  in 
a  single  object  is  Conception.  The  act  is  the  same  for  all 
objects  comprehending  a  number  of  attributes,  or  for  all  terms 


20  ELEMENTS  OF  LOGIC 

comprehending  a  number  of  individual  objects.  Thus  man 
besides  representing  a  whole  which  is  a  combination  of  quali- 
ties, may  denote  a  whole  which  is  a  common  concept  for  a 
class  made  up  of  individual  men.  It  requires,  however,  the 
same  act  of  synthesis  or  comprehension  for  both  kinds  of  con- 
cepts. The  nature  and  properties  of  these  various  concepts 
must  be  examined  later.  We  can  now  only  take  account  of 
the  comparing  and  combining  act  which  groups  different  qual- 
ities or  different  individuals  under  the  same  idea  or  term. 
The  laws  by  which  the  process  is  governed  are  the  laws  of  as- 
sociation, of  identity  and  difference,  and  of  necessary  connec- 
tion. 

Concepts,  as  already  intimated,  are  the  products  of  Concep- 
tion. They  denote  the  ideas  themselves  instead  of  their 
names.  Technically  a  concept  may  be  denned  as  a  synthesis 
of  percepts,  or  a  synthesis  of  individual  wholes  to  form  a  gen- 
eral notion.  If  the  former  could  be  called  a  simple  concept, 
the  latter  could  be  denned  as  the  synthesis  of  simple  concepts, 
making  a  complex  and  general  concept  of  greater  extent.  A 
simple  concept  in  this  view  of  the  case  would  be  such  as 
"man,"  "tree,"  "lion,"  "institute,"  "President,"  etc.,  in  so  far 
as  these  terms  or  ideas  represented  an  individual  group  of 
qualities.  But  in  so  far  as  they  were  general  names  for  a  large 
number  of  individuals  of  the  same  kind,  the  concept  would  be 
complex  as  involving  the  idea  of  a  group  of  qualities  compre- 
hended in  each  member  of  a  larger  group  of  individual  ob- 
jects. But  the  definition  of  it  as  a  synthesis  is  perhaps  too 
technical  for  common  use.  Hence  it  may  be  best  to  define  a 
concept,  in  its  broadest  sense,  as  the  notion  of  a  group  of  qual- 
ities or  individuals  ivhich  are  capable  of  being  thought  of  at  the 
same  time,  the  former  of  which  belong  to  the  same  thing  and 
the  latter  of  which  represent  the  same  general  idea,  or  is  ap- 
plicable to  a  number  of  the  same  kind.  But  this  definition 
will  not  throw  much  light  on  the  subject  because  it  undertakes 
to  do  so  much.  It  is  an  endeavor  to  define  an  idea  which  is 
Hiipposed  to  apply  equally  to  an  individual  group  of  qualities 
and  a  general  group  of  individuals.     The  difference  between 


ELEMENTS  OF  LOGICAL  DOCTRINE  21 

these  two  things  is  so  great  that  some  logicians  use  the 
term  "  concept "  as  if  it  could  apply  but  to  one  of  them  and 
that  to  the  latter,  where  it  expresses  a  general  abstract  idea. 
Some  have  endeavored  to  distinguish  between  "idea,"  "no- 
tion," and  "  concept."  For  certain  purposes  this  may  be 
proper  and  necessary.  A  complete  and  accurate  logical  doc- 
trine may  be  greatly  helped  thereby  inasmuch  as  such  a  dis- 
tinction is  intended  to  evade  the  ambiguities  incident  to  the 
use  of  the  term  "  concept."  But  practical  purposes  may  not 
require  any  refined  niceties  in  the  matter,  and  as  we  are  at 
present  interested  only  in  throwing  light  upon  the  fundamen- 
tal nature  of  Conception  as  a  process,  we  may  postpone  the 
immediate  consideration  of  that  question.  It  will  come  up 
again  when  studying  the  formation  of  concepts  and  their  sev- 
eral divisions.  For  the  moment  we  need  only  to  know  that 
there  is  a  mental  process  by  which  more  than  one  object  of 
consciousness  is  perceived  as  constituting  a  single  wdiole. 
This  process  is  one  of  comparison  and  synthesis.  It  is  ele- 
mentary, and  conditions  all  the  higher  and  more  complex  acts 
of  comparison  and  synthesis.  The  complete  investigation  of 
its  nature  and  functions  maybe  left  either  to  Psychology  or  to 
more  elaborate  treatises  of  Logic  than  the  present  one  pro- 
fesses  to  be.  It  is  sufficient  at  present  to  know  enough  of  it 
for  appreciating  its  relation  to  concepts  and  the  kind  of 
knowledge  with  which  Logic  deals. 

2.  The  Formation  of  Concepts. — I  have  stated  my  intention 
not  to  draw  any  important  distinctions  between  "idea,"  "  no- 
tion," and  "  concept."  For  practical  purposes  I  shall  assume 
them  as  identical  and  as  denoting  objects  of  knowledge  or 
products  of  mental  action  other  than  the  primary  experiences 
of  sense  and  consciousness.  They  may  require  to  be  distin- 
guished from  representative  states  which  are  the  objects  of 
memory.  But  in  a  broader  sense  the  products  of  memory  are 
non-compared  data,  and  may  be  characterized  as  a  form  of 
ideal  percepts.  I  cannot,  however,  go  into  any  special  ques- 
tions of  Psychology.  We  must  be  content  to  assume  that  con- 
cepts are  the  result  of  some  form  of  comparison  and  of  group- 


22  ELEMENTS  OF  LOGIC 

ing  either  qualities  or  objects.  With  this  once  accepted,  at 
least  tentatively,  we  may  proceed  to  see  how  they  have  been 
formed. 

I  said  that  a  concept  might  be  a  synthesis  of  percepts.  But 
the  term  "  percept "  requires  definition  and  suggests  the  ne- 
cessity of  considering  briefly  the  elementary  processes  of 
knowledge. 

A  percept  is  an  individual  object  of  cognition,  such  as  a  color, 
a  sound,  a  taste,  solidity,  weight,  extension.  We  may  include 
any  state  of  self-consciousness  as  such,  and  as  a  single  object 
of  inner  cognition.  The  consciousness  even  of  a  "thought," 
in  the  sense  of  a  logical  act,  so  far  as  it  is  an  individual  ob- 
ject of  inner  perception,  maAr  be  regarded  as  a  percept.  But 
those  illustrations  taken  from  the  phenomena  of  sense  percep- 
tion are  the  clearest  instances,  and  they  suffice  to  show  how  a 
concept  is  formed.  We  must  be  careful,  however,  in  thinking  of 
"  a  percept  as  an  individual  object  of  cognition,"  that  we  do  not 
understand  by  it  any  object  of  sense  perception  in  the  ordinary 
sense  of  the  term,  such  as  a  tree,  a  house,  a  cloud,  matter,  etc. 
We  may  say  practically  that  we  see  such  objects,  but  in  reality 
we  do  not.  We  perceive  certain  colors  and  relations  of  form, 
and  other  experiences  of  the  use  and  nature  of  objects  having 
these  qualities  enable  us  to  interpret  the  meaning  of  what  we 
see.  Thus  what  appears  to  be  a  very  simple  idea  turns  out  to  be 
very  complex.  Hence  a  percept  strictly  considered  can  only  be 
a  single  presentation  of  sense  or  of  consciousness.  Sound  is 
given  by  hearing  only,  savor  by  taste,  solidity  and  resistance 
by  touch,  color  and  form  by  vision.  Certain  signs  also  in 
other  senses  may  be  called  the  percept  by  which  extension 
or  space  is  derived.  But  not  to  go  into  special  problems,  we 
find  these  illustrations  sufficient  to  show  what  is  meant  by  "an 
individual  object  of  cognition."  They  are  the  elements  out  of 
which  concepts  are  formed. 

The  process  of  knowledge  which  gives  us  percepts  is  Per- 
ception or  Apprehension,  which  we  may  denominate  internal 
or  external,  according  as  it  gives  us  material  or  mental  per- 
cepts.    Into  its  nature  Logic  does  not  require  us  to  enter.    It 


ELEMENTS  OF  LOGICAL  DOCTRINE  23 

is  the  process  which  combines  them  that  concerns  the  prob- 
lems of  Logic.  We  call  that  process  Conception,  as  an  act, 
because  it  involves  some  form  of  comparison  and  discrimina- 
tion ;  that  is,  the  holding  of  two  or  more  elements  in  con- 
sciousness at  the  same  time,  and  combining  or  separating  them 
in  its  own  peculiar  way.  But  the  process  is  conditioned  by 
the  Laws  of  Association.  By  repeated  experiences  in  the  con- 
joint perception  of  a  number  of  qualities,  physical  or  mental, 
or  physical  and  mental  combined,  we  come  to  form  a  concept 
of  an  individual  object  or  whole.  By  conjoint  perception  we 
mean  the  activity  of  different  senses  at  the  same  time,  or  un- 
der such  circumstances  as  enables  the  mind  to  think  that  the 
qualities  perceived  belong  to  the  same  thing.  Thus  certain  ex- 
periences of  taste,  color,  and  touch  enable  me  to  form  an  idea 
of  an  orange  as  uniting  the  three  sets  of  qualities,  so  that  when 
I  see  one  of  them  I  may  think  of  the  others  as  associated  with 
it,  and  requiring  only  the  proper  experience  to  verify  them. 
Again,  certain  experiences  of  form,  color,  solidity,  etc.,  and  per- 
ceptions of  usefulness  enable  me  to  form  the  notion  of  a  house. 
I  think  of  these  qualities  as  cohering  in  the  same  object.  The 
same  is  true  of  all  objects  or  wholes  which  we  know.  The  in- 
dividual percepts,  through  the  agency  of  association,  and  per- 
haps other  mental  acts,  are  conceived  as  constituting  one 
object  or  mental  totality,  which  is  a  concept  of  the  first  order. 
All  such  objects  representing  a  simple  synthesis  of  qualities 
or  percepts  I  shall  call  individual  or  attribute-wholes.  The 
simplest  illustrations  of  them  is  that  of  pi-oper  names,  such 
as  Plato,  Bucephalus,  etc.  They  represent  objects  which  are 
merely  a  combination  of  qualities  or  attributes,  and  whose 
name  is  not  applicable  to  any  other  individual.  They  are  only 
attribute-wholes.  Other  and  general  concepts  may  be  more 
than  attribute-wholes.  Thus  "man,"  "quadruped,"  "tree," 
"  circle,"  "  country,"  "  institution,"  "  state,"  etc.,  are  attribute- 
wholes,  inasmuch  as  they  represent  certain  combinations  of 
qualities.  But  they  are  also  more  than  these  at  the  same  time 
because  they  may  denote  a  class  of  individuals.  This  fact 
makes  it  necessary  to  examine  into  the  formation  of  concepts 


24  ELEMENTS  OF  LOGIC 

of  the  second  order.  In  comparison  with  this,  those  of  the 
first  order  may  be  best  denominated  individual-ivholes.  Those 
of  the  second  order  shall  be  called  class-wholes. 

The  process  of  formation  is  that  of  association  and  com- 
parison. If  we  perceive  objects  having  like  qualities  there  is 
a  tendency  to  associate  them  and  to  give  them  a  common  name. 
Thus  I  may  see  an  oak  having  roots,  trunk,  branches,  leaves, 
color,  form,  etc.,  of  a  particular  kind,  and  then  an  elm  with  the 
same  properties  slightly  modified.  But  there  may  be  a  suffi- 
cient number  of  qualities  alike  for  me  to  think  that  the  objects 
bearing  them  are  entitled  to  be  called  by  a  common  name. 
Hence,  instead  of  calling  them  both  "oaks,"  or  both  "elms,"  I 
may  use  the  term  tree,  which  will  apply  to  the  common  quali- 
ties, while  "oak"  and  "elm"  may  denote  the  differences.  We 
may  even  compare  different  "  oak  "  trees  and  use  that  term  as 
a  general  one  to  denote  all  trees  having  the  essential  qualities 
of  "  oak,"  but  with  slight  differences  ;  or  to  denote  all  indivi- 
dual oak-trees  of  exactly  the  same  kind,  whether  there  be  any 
differences  or  not.  The  comparison  is  one  based  upon  the 
similarity  or  resemblance  between  individual  objects,  not  upon 
the  coexistence  of  two  or  more  qualities  in  the  same  thing,  but 
rather  the  coexistence  or  succession  of  the  same  quality  or 
qualities  in  different  things.  The  concept  is  thus  a  class  con- 
cept because  it  applies  to  more  than  one  individual.  It  may 
denote  at  the  same  time  an  individual-whole,  which  is  perhaps 
what  we  represent  to  ourselves  in  imagination  when  we  think 
of  it.  But  it  also  denotes  a  class  of  individuals  to  each  of  which 
it  is  equally  applicable.  The  terms  "man,"  "quadruped," 
"tree,"  "circle,"  " institution, "  "state,"  etc.,  to  which  ref- 
erence was  made  as  representing  attribute-wholes,  are  also 
class-wholes,  or  general  concepts,  as  opposed  to  individual  con- 
cepts, although  they  denote  at  the  same  time  a  totality  of  co- 
hering attributes  or  qualities  which  make  them  representative 
of  individual  concepts.  But  there  are  general  concepts  which 
represent  a  class  of  individuals  without,  perhaps,  indicating 
that  the  individuals  composing  it  are  complex  attribute  wholes. 
These  will  be  such  as  represent  different  percepts,  qualities,  or 


ELEMENTS  OF  LOGICAL  DOCTRINE  25 

objects  of  the  same  kind,  or  perhaps  of  mutually  exclusive 
qualities  not  belonging  to  the  same  object  and  the  same  space. 
Thus  "color,"  "sound,"  "hardness,"  may  be  general  or  class 
concepts  without  denoting  individual-wholes.  Take  the  idea 
of  "  red,"  for  example.  It  is  a  single  percept.  But  this  percept 
may  be  found  in  connection  with  various  objects,  and  so  we 
may  form  a  general  idea  of  "redness"  applicable  to  a  number 
of  percepts  without  denoting  their  synthesis  or  combination 
in  the  same  object  or  the  same  space.  These  will  afford  the 
clearest  illustration  of  class- wholes  independently  of  individual 
wholes,  while  most  concepts  may  be  understood  in  both  rela- 
tions at  the  same  time.  But  in  all,  the  same  process  of  com- 
parison, discrimination,  and  association  is  necessary  to  their 
formation.  The  difference  between  the  two  classes  of  concepts, 
and  henco  between  the  processes  forming  them,  is,  that  indi- 
vidual wholes,  as  a  synthesis  of  percepts,  are  a  combination  of 
different  qualities,  and  class-wholes  are  a  combination  of  similar 
qualities  in  different  objects  under  a  common  name.  To  illus- 
trate again,  "Lincoln,"  "Socrates,"  "Italy,"  "God,"  etc.,  are 
pure  individual  wholes ;  "  color,"  "  sound,"  "  redness,"  "  stature," 
"number,"  etc.,  are  pure  class-wholes,  inasmuch  as  they  de- 
note more  than  one  thing  without  implying  a  union  of  various 
attributes  in  it ;  while  "  man,"  "vegetable,"  "oak,"  "nation," 
etc.,  may  denote  both  individual  and  class  concepts. 

This  distinction  between  the  two  kinds  of  class  concepts 
is  the  basis  of  the  distinction  between  the  Mathematical  or 
Quantitative  Sciences,  and  the  Metaphysical  or  Qualitative 
Sciences.  The  element  of  the  distinction  to  which  I  refer  is 
that  in  which  a  class  term  or  concept  may  be  used  to  denote  a 
number  of  individuals,  perceptive  or  conceptive,  absolutely  iden- 
tical in  k-ind,  or  it  may  denote  a  number  of  individuals  essen- 
tially  different  in  kind,  but  with  common  properties  of  a  char- 
acter to  justify  the  use  of  a  common  name.  The  comparison 
in  one  case  is  between  individual-wholes,  or  properties  to  which 
the  common  name  applies  in  the  same  sense  and  without  a 
difference.  In  the  other  the  comparison  is  between  individual- 
wholes,  or  properties  to  which  the  name  applies  with  a  differ- 


26  ELEMENTS  OF  LOGIC 

ence.  Thus  "  two,"  "  red,"  "  length,"  etc.,  will  apply  to  a  class 
of  objects  without  reference  to  a  difference  in  kind.  These 
may  be  called  mathematical  concepts,  because  each  object  de- 
noted by  them  may  be  representative  of  the  whole  class  irre- 
spective of  the  distinction  between  common  and  accidental,  or 
what  are  called  essential  and  differential  qualities.  But  there 
are  other  concepts,  such  as  "man,"  "tree,"  "biped,"  "Greek," 
"country,"  etc.,  which  may  be  called  logical  or  metaphysical 
concepts,  inasmuch  as  they  denote  individuals  with  such  differ- 
ences that  no  one  of  the  individuals  is  wholly  representative  of 
the  class.  They  are  concepts  connoting  only  the  common  or 
essential  qualities  and  not  taking  any  special  account  of  the 
accidental  or  differential  properties.  It  may  not  be  proper  to 
call  them  "  logical "  or  "  metaphysical,"  and  I  shall  not  contend 
for  that  name  longer  than  to  indicate  the  distinction  which  I 
have  defined.  The  importance  of  the  distinction  itself  will 
appear  when  we  discuss  the  matter  of  genus  or  essentia,  and 
differentia  or  accident.  We  require  at  present  only  to  recog- 
nize the  two  acts  of  comparison,  one  of  which  involves  an  act 
of  abstraction,  and  the  other  does  not.  The  formation  of  math- 
ematical concepts  does  not  require  any  abstraction  of  special 
properties,  and  the  bringing  of  them  together  to  the  neglect  of 
certain  others,  unless  it  be  in  a  manner  which  we  hardly  need 
to  take  account  of.  On  the  contrary,  the  formation  of  what  I 
have  called  logical  or  metaphysical  concepts,  such  as  "  man," 
"tribe,"  "race,"  "animal,"  etc.,  requires  that  we  abstract  cer- 
tain common  qualities  and  ignore  the  accidental  ones,  so  as  to 
denote  the  former  by  a  common  name.  "  Abstraction  "  is  here 
used  merely  to  denote  the  thinking  away  of  certain  properties 
from  their  exclusive  application  to  any  particular  object.  A 
farther  conception  of  it  is  not  required  at  present.  It  is  suffi- 
cient to  remark  a  process  producing  two  kinds  of  class  con- 
cepts with  which  later  problems  in  logic  will  have  to  reckon. 

3.  The  Denomination  of  Concepts. — The  naming  of  con- 
cepts has  always  been  considered  an  important  matter  in 
Logic.  It  is  not  especially  a  process  of  thought,  although  it  is 
necessary  to  make  thought  an  effective  instrument  in  the  com- 


ELEMENTS  OF  LOGICAL  DOCTRINE  27 

niunication  of  knowledge,  and  to  give  stability  and  fixity  to  the 
various  products  of  mental  activity.  Some  have  even  consid- 
ered names  and  naming  the  most  important  aspects  of  Logic. 
Into  this  controversy  it  is  not  necessary  to  enter  either  on  one 
side  or  the  other.  We  may  be  content  with  one  or  two  re- 
marks about  the  nature  of  language  and  its  service  to  thought. 
Language  is  the  symbol  of  ideas,  or  consists  of  the  signs  by 
which  we  can  indicate  the  resemblances  and  differences  be- 
tween concepts.  By  it  the  infinite  number  of  individuals  and 
classes  can,  to  some  extent,  be  tabulated  or  indexed  for  use. 
Without  it,  perhaps,  we  should  not  be  able  to  develop  our 
thinking  processes  above  the  grade  of  animal  intelligence. 
With  it  we  can  name  an  idea  so  as  to  keep  it  by  itself,  if  re- 
quired, or  conceive  and  speak  of  a  class  of  ideas,  if  need  be. 
The  same  word  even  may  have  a  double  denotation,  as  we  have 
seen  in  general  concepts  denoting  either  individual  or  class- 
wholes.  In  this  way  a  word  may  indicate  the  quality  that  sep- 
arates the  concept  or  object  named  by  it  froni  others,  or  it 
may  apply  equally  to  all  members  of  the  class.  "  Man  "  may 
imply  or  represent  the  quality  or  qualities  which  separate  him 
from  a  "lion,"  an  "eagle,"  or  all  other  objects.  At  the  same 
time  it  will  denote  the  qualities  by  which  the  term  may  be 
employed  to  indicate  all  individual  men.  Thus  economy  of 
language  is  obtained,  on  the  one  hand,  and  clearness  of  con- 
ception on  the  other.  But  the  general  service  of  language  is  il- 
lustrated in  the  simple  fact  that  where  any  ambiguity  is  asso- 
ciated with  a  single  term,  our  intellectual  confusion  is  com- 
pletely overcome  by  the  use  of  two  or  more  terms  which  may 
specify  the  distinct  qualities  confounded  under  a  single  term. 
The  denomination  of  concepts,  therefore,  is  an  important  pro- 
cess either  in  completing  the  act  of  thought,  or  in  making  it 
useful  when  it  is  completed.* 

*  The  student  may  consult  the  following  references  for  a  discussion 
of  Language  and  Denomination  in  their  relation  to  Logic  :  Bosanquet  : 
Logic,  vol.  i.,  Introduction.  Sections,  4,  5,  pp.  8-30;  Thomson:  Laws 
of  Thought,  Introduction,  Chapter  on  Language  ;  Hamilton  :  Lectures  on 
Logic,  Lecture  VIII. 


28  ELEMENTS  OF  LOGIC 

2d.  Judgment. — Like  Conception  the  term  Judgment  may 
denote  both  a  process  and  a  product.  Its  product  is  a  propo- 
sition. At  present  we  are  concerned  only  with  the  act,  which 
can  be  briefly  defined,  and  which,  with  a  slight  modification, 
will  apply  to  the  product.  As  an  act  a  judgment  is  the  per- 
ception of  the  relation  between  two  concepts.  It  is  an  act  of 
comparison  still  more  distinct  than  any  involved  in  Concep- 
tion. The  relation  which  it  expresses  is  that  of  agreement  or 
difference  between  subject  and  predicate,  or  of  inhesion  or 
non-inhesion  between  subject  and  attribute.  Much  discussion 
would  be  recpiired  to  give  a  satisfactory  theory  of  that  rela- 
tion, and  to  decide  whether  it  is  wholly  one  of  agreement  or 
difference,  or  partly  of  some  other  character.  But  larger 
treatises  may  be  consulted  upon  this  problem.  It  is  sufficient 
to  know  that  the  act  of  connecting  two  concepts  is  the  funda- 
mental characteristic  of  Judgment.  The  laws  which  determine 
it  may  be  ascertained  without  any  particular  theory  of  the 
process,  even  if  such  a  theory  be  helpful  in  the  final  solution 
of  problems  centring  about  it. 

The  principal  matter  of  importance  to  be  observed  in  con- 
nection with  the  nature  of  Judgment  is  that  the  act  of  compar- 
ison involved  in  it  is  in  its  essential  elements  the  same  as  the 
act  of  Conception.  The  difference  between  them  is  only  the 
way  in  which  the  result  is  expressed,  or  the  object-matter  with 
which  they  deal.  Conception  is  the  connecting  of  percepts ; 
Judgment  is  the  connecting  of  concepts.  The  connecting  of 
percepts  can  be  expressed  in  terms  or  words :  the  connecting 
of  concepts  must  be  expressed  in  propositions.  The  relation 
between  subject  and  predicate  in  Judgment  may  be  variously 
expressed.  But  the  process  will  be  identical  with  that  which 
is  involved  in  the  formation  of  concepts.  This  is  perhaps  ap- 
parent in  the  fa<;t  that  the  two  kinds  of  judgments  correspond 
to  the  two  general  kinds  of  concepts.  There  are  judgments 
which  express  the  relation  between  substance  and  attribute, 
corresponding  to  concepts  representing  individual  wholes  ;  for 
example,  "  Iron  is  hard."  Then  there  are  judgments  which 
express  the  relation  of  resemblance  or  difference  between  sub- 


ELEMENTS  OF  LOGICAL  DOCTRINE  29 

ject  and  predicate,  or  the  relation  of  classes,  corresponding  to 
concepts  representing  class-wholes  ;  for  example,  "Iron  is  a 
metal."  We  shall  examine  the  importance  of  this  distinction 
again.  The  nature  of  the  mental  process  of  Judgment  and  its 
relation  to  that  of  Conception  is  all  we  require  to  know  at  this 
stage  of  the  discussion. 

3d.  Reasoning. — Reasoning  is  a  process  only  a  little 
more  complex  than  Judgment.  It  may  be  briefly  defined  as 
inference.  But  this  will  require  further  explanation.  Hence 
we  may  adopt  Jevons's  definition  of  reasoning  as  adequate  for 
all  practical  purposes.  It  is  "  the  progress  of  the  mind  from 
one  or  more  given  propositions  to  a  proposition  different  from 
those  given.  Those  propositions  from  which  we  argue  are 
called  the  Premises,  and  that  which  is  drawn  from  them  is 
called  the  Conclusion."  The  definition  here  covers  what  is 
known  as  Immediate  Inference,  and  Mediate  Inference.  If 
from  the  proposition  that  "All  metals  are  elements,"  I  infer 
that  "All  that  are  not  elements  are  not  metals,"  I  am  mak- 
ing an  immediate  inference  ;  if  from  the  two  propositions 
that  "All  metals  are  elements,"  and  "Iron  is  a  metal,"  I  in- 
fer that  "  Iron  is  an  element,"  I  am  making  a  mediate  infer- 
ence. But  in  both  cases  my  reasoning  consists  in  the  percep- 
tion of  a  relation  between  pi^ositions,  through  the  medium 
or  agency  of  concepts  whose  relation  is  known,  implied,  or  ex- 
pressed. 

This  last  definition  of  Reasoning  identifies  the  process  in  its 
nature  with  that  of  Conception  and  Judgment.  The  difference 
between  them  is  not  in  the  form  of  the  mental  act,  but  in  the 
matter  to  which  it  is  applied.  In  reasoning  it  is  the  resem- 
blance or  difference  of  relations  between  propositions  that  con- 
stitutes the  peculiar  nature  of  the  matter  dealt  with,  while  in 
Judgment  it  is  the  resemblance  or  difference  of  relations  be- 
tween concepts.  The  distinction  is  thus  only  a  matter  of 
complexity,  Reasoning  representing  in  a  more  complex  form 
only  what  is  found  substantially  in  the  earlier  process  of  Con- 
ception. At  any  rate,  this  way  of  regarding  the  question  helps 
to  give  unity   to  the  mental  processes,  while  it  justifies  our 


30  ELEMENTS  OF  LOGIC 

turning  to  the  differences  of  matter  to  which  the  various  men- 
tal acts  are  applied,  in  order  that  we  may  determine  the  differ- 
ent laws  of  thought  regulating  the  process  according  to  the 
changes  of  matter  apparent  in  the  development  of  the  subject 
of  Logic. 


CHAPTER  m. 

TERMS  OR  CONCEPTS,   AND  THEIR  KINDS 

Terms  and  concepts  have  already  been  denned  as  denoting 
ideas.  We  come  next  to  their  divisions,  which  can  be  made 
on  several  different  principles  of  distinction. 

1st.  Categorematic  and  Syncategorematic  Terms. — 
A  categorematic  word  is  one  which  can  stand  as  the  subject 
or  predicate  of  a  proposition.  Such  are  "animal"  "nation," 
"excellence,"  "wise,"  "beautiful,"  "perfect."  These  show 
that  categorematic  terms  are  limited  to  nouns  and  adjectives. 
Verbs  ought  to  be  included. 

A  syncategorematic  word  is  one  which  cannot  stand  alone 
as  the  subject  or  predicate  of  a  proposition.  Such  are  "  with," 
"  and,"  "  through,"  "nobly,"  "  very,"  " indeed,"  etc.  From  this 
we  perceive  that  syncategorematic  terms  are  either  modal  or 
relational :  modal,  if  they  are  adverbs,  relational,  if  they  are 
conjunctions  and  prepositions. 

A  term  in  the  logical  sense,  as  the  subject  or  predicate 
of  a  proposition,  may  consist  of  one  or  more  categorematic 
words,  or  of  categorematic  and  syncategorematic  words.  In 
the  latter  case  it  must  consist  of  a  grammatical  phrase  or 
clause. 

2d.  Singular  and  Ceneral  Terms. — This  division  of 
terms  is  limited  to  categorematic  words,  and  perhaps  to  the 
class  of  substantives.  But  this  fact  is  less  important  than  that 
the  division  is  a  new  one  and  not  to  be  confused  with  the  one 
already  made.  It  is  based  upon  the  differences  between  the 
number  of  individuals  denoted  by  various  concepts,  and  at 
least  partly  coincides  with  that  between  individual-wholes  and 
class-wholes. 

1.  Singular  Terms. — A  singular  term  is  one  which  can  be 


32  ELEMENTS  OF  LOGIC 

affirmed,  in  the  same  sense,  only  of  a  single  object,  real  or  im- 
aginary. Proper  names  are  the  best  illustration.  Thus,  Na- 
poleon, Paris,  Greece,  St.  Paul,  etc.,  are  singular  terms  because 
they  can  aj^ply,  in  the  same  sense,  only  to  one  object.  Ex- 
pressions like  "the  Secretary  of  State,"  "the  Prime  Minister 
of  England,"  "  the  King  of  Spain,"  will  be  singular  when  they 
refer  to  an  individual  or  particular  person.  But  they  are  also 
capable  of  being  general  terms.  This  will  be  the  case  when 
they  are  used  to  denote  the  class  of  officers  by  those  respec- 
tive names.  Thus  when  we  refer  indefinitely  to  "  the  Secretary 
of  State  "  as  any  man  or  officer  in  that  station,  we  use  it  in  its 
general  sense  ;  if  we  refer  definitely  to  a  particular  man  occu- 
pying the  place,  it  is  singular.  Keynes  also  indicates  how  the 
same  expressions  may  become  singular,  if  an  individualizing 
prefix  is  added  to  them  :  Thus,  "  the  present  Prime  Minister," 
"  the  present  Secretary  of  State,"  "  the  reigning  Queen  of 
England,"  etc.  Likewise,  he  thinks  such  expressions  as  "the 
first  man,"  "  the  pole  star,"  are  singular.  Perhaps  we  could 
add  such  terms  as  "  the  highest  good,"  "  the  supreme  or  ulti- 
mate end."  These  are  simple  cases  where  the  addition  to  a 
general  name  of  an  adequate  prefix  transforms  it  into  a  sin- 
gular one.  But  great  caution  must  be  exercised  in  our  judg- 
ment of  such  cases.  For  example,  "  the  eldest  child,"  is  an  ex- 
pression which  is  applicable  only  to  a  particular  child,  but  it 
is  an  indefinite  particular,  and  so  is  general  in  its  import.  But 
the  illustrations  preceding  show  that  there  may  be  singular 
terms  other  than  proper  names.  Some  of  them,  however,  are 
capable  of  both  a  general  and  a  singular  use.  For  example, 
sjmce,  when  it  refers  to  the  totality  by  that  name,  is  singu- 
lar, but  when  it  is  a  name  for  a  number  of  definite  spaces, 
or  the  different  portions  of  aggregate  space,  it  has  all  the 
distinctive  characters  of  a  general  term.  The  same  is  true 
of  the  term  universe,  and  Professor  Bain  thinks  it  true  of 
all  aggregate  substances  which  are  divided  into  parts,  not 
kinds;  as  "water,"  "stone,"  "salt,"  "mercury,"  "flame." 
But  Keynes's  remark  that  we  can  say  "some  water,"  "some 
salt,"  "some  mercury,"  which  cannot  be  said  of  proper  names 


TERMS  OR  CONCEPTS,   AND   THEIR  KINDS         33 

or  concepts  denoting  only  one  object,  or  an  individual  whole, 
seems  to  me  decisive  in  favor  of  considering  them  general 
terms.  Keynes  ought,  perhaps,  to  have  remarked  the  reason 
that  such  terms  are  liable  to  be  mistaken  for  singular  ones. 
It  is  that  general  terms  apply  either  to  objects  which  are  dif- 
ferent in  kind,  or  to  objects  which  have  an  independent  exist- 
ence, and  never  unite  to  form  one  continuous  mass  of  homo- 
geneous matter,  as  do  "water,"  "air,"  "space,"  and  "stone" 
when  denoting  rock  of  the  same  kind.  Oneness  of  kind  is  a 
characteristic  of  singular  terms,  as  well  as  individuality,  and 
hence  it  is  easy  to  confuse  such  terms  as  "water,"  "air," 
"  mercury,"  etc.,  with  the  singularity  of  proper  names.  But 
they  represent  objects  which  can  be  divided  into  individual 
parts  without  modifying  the  qualities  of  the  parts  which  re- 
ceive the  same  names  as  the  aggregates.  This  is  not  true  of 
proper  names,  or  of  any  singular  temi  which  is  singular  in  its 
absolute  sense.  It  would  have  been  more  apt  if  Professor 
Bain  had  classified  the  terms  as  collective,  which  are  still  to 
be  examined,  for  their  analogy  to  such  terms  is  closer  than  to 
singular  terms.  But,  nevertheless,  I  think  they  can  be  shown 
not  to  be  collective. 

Proper  names  may  become  general  terms  when  used  to  de- 
note a  class  of  individuals  having  a  given  characteristic.  Thus 
"the  Napoleons,"  "the  Csosars,"  "the  Platos,"  "the  Washing- 
tons,"  are  general  terms  because  they  denote  a  class  of  persons 
with  certain  characteristics  which  belonged  to  the  original 
person  by  that  name.  Again,  the  name  "  God  "  will  be  singu- 
lar to  a  monotheist  and  general  to  a  polytheist :  uncajutalized 
it  is  general  to  a  monotheist. 

There  is,  then,  no  absolute  rule  by  which  the  mere  form  of  a 
word  may  be  taken  to  indicate  its  character.  Some  general 
terms  by  adding  a  prefix  may  become  singular  ;  some  singular 
terms,  used  in  the  plural,  or  to  denote,  on  certain  emergencies, 
more  than  one  of  a  kind,  become  general.  It  is  a  case  where 
a  change  of  the  matter,  when  the  form  of  the  term  remains 
absolutely  or  virtually  the  same,  affects  the  character  of  the 
concept.  Hence  the  ride  can  be  absolute  only  when  the  two 
3 


34  ELEMENTS  OF  LOGIC 

aspects  of  the  concept,  form  and  matter,  remain  constant  and 
according  to  definition. 

2.  General  Terms. — A  general  term  or  concept  is  one  which 
can  be  applied,  in  the  same  sense,  to  an  indefinite  number  of 
objects,  real  or  imaginary.  It  is  a  name  applied  to  class- 
wholes,  such  as  "man,"  "vertebrate,"  "quadruped,"  "genera- 
tion," "triangle,"  etc.  Also  such  terms  as  "army,"  "forest," 
"crowd,"  "nation,"  etc.,  are  general  terms.  But  there  is  so 
marked  a  difference  between  the  first  class  of  illustrations  and 
the  second  that  a  subdivision  of  general  terms  into  distributive 
and  collective  has  to  be  recognized.  A  distributive  term  is 
one  which  applies  in  the  same  sense  to  each  individual  in  the 
class,  such  as  in  the  first  set  of  illustrations  given  ;  namely, 
"  man,"  "  vertebrate,"  etc.  A  collective  term  is  one  which 
applies  to  an  aggregate  of  individuals  composing  a  whole,  and 
which  will  not  apply  to  any  of  the  individuals  alone.  Thus 
"army,"  "flock,"  "bevy,"  "family,"  "tribe,"  etc.,  are  collective 
terms  because  they  denote  a  composite  or  aggregate  whole. 
They  are  at  the  same  time  general  terms  because  they  are 
applicable  in  the  same  sense  to  an  indefinite  number  of  such 
aggregates. 

It  is  important  to  remark  that  a  collective  term  may  be 
singular  instead  of  general.  Thus  the  Vatican  Library,  the 
American  people,  the  Greek  nation,  the  Seventy-second  Regi- 
ment, Company  B,  etc.,  are  singular  aggregates  because  the 
name  will  apply  to  no  other  objects  of  the  same  general  kind. 
Some  few  terms  may  be  used  either  distributively  or  collective- 
ly according  to  the  emergency.  Thus  "people,"  "the  Greeks," 
"  the  English,"  and  often  the  plural  of  ordinary  general  terms, 
as  "the  trees,"  "the  houses,"  etc.,  may  be  used  in  either  sense, 
as  illustration  will  presently  show.  The  following  diagram 
will  show  the  relation  subsisting  between  singular  and  general 
terms,  and  their  subdivision  : 

i  a-        i       \  t    a-    -j      i  o-        i      i  Individual. 

Terms  J       *         *  ^vidual.      ^    Singular  {  Collectiye 

J  r,  ,     (  Collective.  '     -,  ,  \  Collective.  C 

(  General    ■{  ^..  ,  .,    ..  General-;  r*-  *  -\    *■ 

v  Distributive.  Distributive. 


TERMS  OR  CONCEPTS,   AND   TIIEIR  KINDS         35 

Perhaps  a  doubt  about  the  accuracy  of  this  representation 
is  possible,  since  a  collective  term  when  general  may  at  the 
same  time  have  the  characteristics  of  a  distributive  term. 
This  is  true  of  such  terms  as  "army,"  "herd,"  "regiment," 
etc.  They  are  collective  in  so  far  as  they  denote  an  aggregate 
whole  of  individuals  ;  they  are  distributive  in  so  far  as  they 
denote  or  can  apply  to  a  class  of  such  aggregates.  But  since 
the  last  is  true  only  when  the  aggregates  are  treated  as  indi- 
vidual-wholes, the  general  distinction  between  them  may  still 
be  observed.  It  is  merely  a  case  again  where  the  difference  is 
due  to  some  variation  between  the  form  and  matter  of  the 
concept.  Thus  in  the  sentence,  "  Mobs  are  crowds  of  enraged 
men,"  there  can  be  no  distinction  between  the  form  and  mat- 
ter of  the  collective  term  "  mobs."  But  in  the  sentence,  "  Mobs 
are  dangerous,"  the  assertion  is  made  of  all  such  aggregates, 
and  hence  while  the  form  of  the  term  is  collective  its  matter 
may  be  both  collective  and  distributive,  or  distributive  alone. 

But  the  distinction  between  distributive  and  collective 
terms  is  more  important  for  Logic  than  the  distinction  be- 
tween singular  and  general  terms.  This  is  apparent  for  the 
reason  that  the  nature  of  propositions  is  less  affected  by  the 
latter  than  by  the  former  distinctions.  We  shall  learn  later 
that  the  same  logical  laws  aj^ply  to  "singular  propositions" 
as  apply  to  "universal  propositions,"  and  the  distinction  be- 
tween these  is  parallel  with,  and  determined  by,  that  between 
singular  and  general  terms.  They  are  not  liable  to  easy  con- 
fusion. But  the  collective  and  distributive  uses  of  terms  are 
often  confused  and  give  rise  to  corresponding  fallacies  in  rea- 
soning. This  liability  to  confusion  is  illustrated  in  such 
propositions  as  the  following  :  "  All  the  angles  of  a  triangle 
are  equal  to  two  right-angles,"  and  "  All  the  angles  of  a  tri- 
angle are  less  than  two  right-angles."  The  two  propositions 
seem  contradictory,  because  the  same  thing  cannot  be  equal 
to  and  less  than  another  at  the  same  time  and  taken  in  the 
same  sense.  But  the  first  proposition  taken  collectively  in 
its  subject  is  true,  and  the  second  taken  distributively  is  true. 
Again,  "All  the  trees  in  the  forest  produce  a  thick  shade," 


36  ELEMENTS  OF  LOGIC 

may  be  taken  in  the  same  double  sense.  Similarly,  "  The  peo- 
ple filled  the  hall,"  and  "  The  people  are  honest,"  or,  "  The 
Greeks  are  a  nation,"  and  "  The  Greeks  are  Caucasian."  A 
case  of  the  plural  of  an  ordinary  distributive  term  becoming 
collective  is  the  following  :  "  The  trees  make  a  forest ; "  but  in 
the  following  it  is  distributive  :  "  The  trees  are  beautiful." 

3d.  Concrete  and  Abstract  Terms. — It  is  difficult  to 
give  a  satisfactory  definition  of  concrete  and  abstract  terms. 
Scarcely  any  two  authors  agree  upon  the  subject,  and  even  if 
they  did  agree,  observation  would  teach  us  that  any  attempt 
to  apply  the  definition  to  an  actual  classification  of  concepts 
would  meet  with  the  serious  obstacle  that  the  two  classes  seem 
to  shade  off  into  each  other  by  insensible  degrees.  It  is  only  in 
certain  cases  that  the  distinction  can  be  made  clear,  and  often, 
in  spite  of  this  clearness,  a  term  may  be  concrete  in  one  of  its 
applications  and  abstract  in  another.  The  difficulty  is  largely 
caused  by  the  unfixed  and  varied  use  of  the  term  "  abstract," 
which  is  sometimes  used  as  the  equivalent  of  "  general,"  and 
again  as  denoting  the  conception  of  a  quality  apart  from  its 
subject.  The  confusion  occasioned  by  this  usage  will  appear 
as  we  proceed. 

The  definition  of  "concrete"  and  "abstract"  terms  can  be 
made  very  simjile.  The  only  difficulty  we  have  to  encounter 
after  that,  is  the  determination  of  the  particular  terms  which 
fall  under  the  one  class  or  the  other.  But  we  can  at  least 
begin  with  a  definition  of  them  in  their  purest  form,  and  if 
subsequent  facts  require  us  to  qualify  it  we  can  do  so. 

A  concrete  term  is  a  name  which  stands  for  a  thing,  or  for 
an  attribute  of  a  thing  conceived  as  an  attribute  ;  e.g.,  "Web- 
ster," "Bucephalus,"  "Parthenon,"  "white,"  "clear,"  "gener- 
ous," etc.  A  concrete  concept  is  the  same  in  its  meaning,  but 
we  do  not  think  of  it  as  a  word.  It  denotes  a  tiling  or  a  qual- 
ity as  the  object  of  consciousness,  while  spoken  of  as  a  term 
it  is  the  object  of  Grammar.  With  the  same  qualification  we 
may  define  abstract  terms  or  concepts. 

An  abstract  term  is  a  name  which  stands  for  an  attribute  or 
quality  considered  apart  from  the  thing  possessing  it,  or  con- 


TERMS  OR  CONCEPTS,   AND   THEIR  KINDS         37 

ceived  and  used  as  a  thing,  e.g.,  "  redness,"  "  cheapness," 
"purity,"  "perfection,"  "righteousness,"  "justice,"  "ability," 
etc. 

Mill's  definition  is  as  follows  :  "  A  concrete  name  is  a  name 
which  stands  for  a  thing  ;  an  abstract  name  is  a  name  which 
stands  for  an  attribute  of  a  thing."  It  is  not  clear  what  such 
a  definition  will  do  with  adjectives.  They  are  names  of  attri- 
butes and  yet  Mill  includes  them  among  concrete  names.  On 
the  other  hand,  they  can  hardly  be  iucluded  among  concrete 
terms  because  they  do  not  denote  "  things  "  in  the  strict  sense. 
We  must  therefore  either  make  a  class  for  them  apart  from 
the  concrete  and  abstract,  or  modify  the  definition.  I  have 
attempted  to  overcome  this  difficulty  in  the  definition  which 
I  have  given.  It  provides  for  two  kinds  of  concrete  terms, 
the  substantive  and  the  attributive.  I  shall  also  divide  abstract 
terms  into  two  classes,  static  and  dynamic,  or  adjectival  and 
verbal  nouns.  To  complete  the  classification  I  shall  set  aside 
a  third  class  of  mixed  concrete  and  abstract  terms.  Thus  it 
appears  in  the  following  table  : 


Terms  < 


r  p  S  Substantive  =  Singular  Xouns,  e.g.,  Homer. 

\  concrete  -(  Attributive  _  Adjectives,  e.g.,  Pure. 
111  e  ')  . ,  j  Static  =  Adjectival  Nouns,  e.g. ,  Generosity. 

{  j   js  rac     |  Dynamic  =  Verbal  Nouns,  e.g.,  Acceleration. 
Mixed  =  Concrete    and    Abstract,    e.g.,    Government,    Institu- 
tion, etc. 


In  this  classification  no  mention  of  such  terms  as  "  man," 
"  animal,"  "race,"  "  vegetable,"  "  triangle,"  etc.,  has  been  made. 
Keynes  considers  them  as  concrete.  The  illustrations  of  con- 
crete substantives  were  chosen  from  singular  terms,  and  it  is 
now  a  question  whether  general  terms  are  concrete  or  abstract. 
The  mere  fact  of  being  general  terms  does  not  make  them 
concrete,  as  is  shown  in  verbal  abstracts,  which  may  be  general, 
and  also  in  the  adjectival  abstracts,  if  they  can  be  consider- 
ed as  general.  Jevons  thinks  them  singular.  This  is  ex- 
tremely doubtful,  to  say  the  least.  But  "  man,"  "  animal," 
"  race,"  etc.,  denote  objects  which  we  are  accustomed  to  speak 
and  think  of  as  "concrete,"  and  so  they  seem  most  naturally 


38  ELEMENTS  OF  LOGIC 

entitled  to  be  called  concrete  names.  On  the  other  hand, 
some  writers  call  them  abstract,  and  so  regard  all  general  con- 
cepts as  abstract  because  the  process  of  forming  them  is  one 
of  abstraction.  Keynes  disputes  the  legitimacy  of  this  treat- 
ment of  them.  He  seems  to  hold  that  the  process  has  nothing 
to  do  with  the  nature  of  the  product.  This  seems  to  my  mind 
doubtful.  But  in  so  far  as  such  terms  as  "man,"  "  animal," 
"  tree,"  etc.,  denote  real  objects  which  are  definite  things,  they 
may  with,  at  least,  tolerable  propriety,  be  called  concrete. 
But  in  so  far  as  they  are  indefinite  and  do  not  seem  to  repre- 
sent any  clearly  conceived  individual  object,  they  closely  re- 
semble abstract  terms  in  this  respect.  In  fact,  as  terms  be- 
come more  general,  that  is,  as  their  extension  increases  they 
approximate  in  indefiniteness  the  abstract  concepts  which  are 
characterized  by  this  indifference  to  particular  things,  and 
hence  they  may  with  some  propriety  be  spoken  of  as  abstract. 
Perhaps  their  ambiguity  in  this  respect  would  justify  us  in 
regarding  them  as  abstract  in  one  relation  and  concrete  in 
another,  their  abstractness  depending  upon  the  proportion  of 
indefiniteness,  and  their  concreteness  upon  the  proportion  of 
definiteness  expressed  by  them.  There  are  other  terms  which 
give  still  greater  trouble  than  these.  They  are  such  examples 
as  "color,"  "sound,"  "pleasure,"  "thought,"  etc.  It  may  be 
a  problem  to  determine  whether  they  are  concrete  or  abstract. 
On  the  one  hand,  "  color  "  and  "  sound,"  as  denoting  quali- 
ties apart  from  a  definite  conception  of  their  subject,  might  be 
regarded  as  abstract.  On  the  other  hand,  as  general  terms 
for  attributes  which  are  concrete  they  might  be  considered  as 
concrete.  So  "pleasure,"  "thought,"  and  all  names  of  states 
of  consciousness,  as  verbal  nouns,  dynamic  concepts,  might  be 
abstract.  But  as  names  of  individual  facts  clearly  represent- 
able  in  some  way,  they  might  be  considered  by  many  writers 
as  concrete.  In  other  words,  as  general  terms  denoting  facts 
or  individual  events,  they  will  be  usually  conceived  as  con- 
crete, but  as  terms  denoting  attributes,  but  not  definitely  de- 
noting their  subject,  they  will  appear  usually  as  abstract. 
This  may  be  true  of  a  large  number  of  terms.     If  so,  they  may 


TERMS   OR   CONCEPTS,    AND    THEIR  KINDS  39 

be  treated  in  two  relations  at  the  same  time,  according  to  the 
degree  of  detiniteness  and  indefiniteness  with  which  they  are 
conceived. 

But  this  manner  of  speaking  about  such  terms  suggests 
the  common  usage  of  the  words  "  concrete  "  and  "  abstract," 
which  logical  discussion  cannot  wholly  ignore.  Ordinarily 
"  concrete "  denotes  any  sensible  or  real  object,  which  may 
represent  the  meaning  of  a  term,  singular  or  general.  Thus 
"  man,"  "  tree,"  "  biped,"  denote  sensible  objects.  Even 
"  color,"  "  sound,"  "  odors,"  denote  individual  percepts,  and  so 
there  is  always  some  distinct  or  definite  reality  expressed  by 
them,  which  is  supposed  to  be  identical  with  the  concrete. 
But  such  terms  as  "emotion,"  "reasoning,"  "thought,"  "pleas- 
ure," "figure,"  "form,"  are  either  not  sensible  objects  or  are 
so  vague  and  indefinite,  being  often  called  the  "  higher  ab- 
stractions "  of  the  mind,  that  the  common  consciousness  thinks 
and  speaks  of  them,  perhaps  loosely,  as  "abstract"  concep- 
tions. In  the  same  way  general  terms  are  often  conceived  as 
abstract  in  proportion  to  their  generalized  character,  or  their 
remoteness  from  the  individuals  which  they  comprehend. 
This  distinction,  then,  is  mainly  that  between  what  is  presenta- 
tive  or  representative,  and  what  is  merely  thought  in  conscious- 
ness. Conceptions  which  call  up  distinctly  to  the  mind  the 
individual  objects  which  they  denote  are  thus  commonly  taken 
for  concrete,  and  those  which  do  not  indicate  clearly  the  char- 
acteristics named,  and  which  might  be  called  symbolical  con- 
ceptions, after  the  manner  of  Leibnitz,  are  taken  as  abstract. 
For  this  reason  it  might  serve  the  purposes  of  Logic  much 
better  if  the  distinction  wrere  made  between  definite  and  indefi- 
nite concepts,  the  former  taken  to  denote  all  terms  which 
clearly  denote  or  connote  certain  marks,  such  as  Peter,  man, 
white,  and  the  latter  taken  to  denote  such  as  do  not  indicate 
distinctly  any  communicable  mark  of  an  object  or  a  fact,  as 
"life,"  "organic,"  "humanity,"  "  government,"  "  institution," 
etc.  It  is  this  distinction  rather  than  that  between  the  con- 
crete and  the  abstract,  as  I  have  defined  them,  that  is  of  im- 
portance in  Logic,  because  fallacious  reasoning  is  occasioned 


40  ELEMENTS  OF  LOGIC 

more  by  ambiguities  of  meaning  due  to  indefiniteness  than  by 
the  question  whether  a  term  is  abstract  or  not,  unless  "  ab- 
tract "  is  taken  as  synonymous  with  indefinite.  As  this  indef- 
initeness increases  with  the  extension  of  a  concept,  general 
terms  will  partake  of  this  character  as  they  recede  from  the 
individual  and  concrete,  and  so  will  often  be  called  "  abstract " 
in  the  same  degree,  although  they  may  not  wholly  lose  a  con- 
crete reference.  It  is  possible,  therefore,  to  consider  them 
either  as  mixed  concrete  and  abstract  terms,  or  as  combining 
definite  and  indefinite  characteristics  in  an  inverse  ratio  to 
each  other.  If  the  former,  we  simply  adapt  them  to  our  defi- 
nitions ;  if  the  latter,  we  use  "concrete"  and  "abstract"  to 
denote  the  distinction  between  the  representable  and  the  non- 
representable  concepts,  which  is  the  most  important  for  Logic, 
as  later  chapters  will  show.  Wundt  has  some  excellent  obser- 
vations upon  this  question,  which  sustain  the  j)Osition  I  have 
taken,  and  should  be  quoted  in  this  connection  : 

"  It  is  a  necessary  consequence  of  the  formation  of  concepts 
from  the  connection  of  percepts  that  certain  conceptions  • 
should  stand  much  nearer  than  others  to  the  presentations  of 
sense.  We  have  generally  expressed  this  fact  by  the  distinc- 
tion between  concrete  and  abstract  ideas,  but  have  described  the 
logical  j>rocess,  by  which  abstract  notions  have  been  formed 
from  concrete,  as  the  process  of  abstraction.  The  influence  of 
this  latter  procedure,  moreover,  has  essentially  changed  the 
meaning  of  the  terms  '  concrete '  and  '  abstract '  in  the 
course  of  history.  Scholastic  Nominalism,  which  introduced 
it  into  Logic,  used  the  terms  for  the  mere  purpose  of  distin- 
guishing between  words.  Every  substantive  noun  which  de- 
noted an  individual  object,  or  a  class  of  objects,  was  concrete, 
whereas,  on  the  other  hand,  a  word  formed  from  a  concrete 
term  and  used  to  denote  a  universal  property  was  called  ab- 
stract. Words  like  man,  tvhite,  etc.,  therefore  were  regarded  as 
concrete,  and  those  like  humanity,  whiteness,  etc.,  were  regarded 
as  abstract.  In  modern  Logic  this  distinction  gradually  became 
confused  with  that  between  individual  and  general  concepts, 
since,  in  depending  upon  the  use  of  the  word  '  to  abstract,' 


TERMS  OR  CONCEPTS,   AND   THEIR  KINDS         41 

we  became  accustomed  to  characterize  as  abstract  all  concep- 
tions whose  formation  was  marked  by  a  distinct  process  of 
abstraction.  But  as  this  was  peculiar  to  all  generic  concep- 
tions, there  remained  finally  nothing  but  singular  or  individual 
terms  which  could  represent  the  territory  of  the  concrete. 
This  confusion  is,  in  fact,  logically  quite  excusable  ;  for,  in 
whatever  senses  the  terms  '  concrete '  and  '  abstract '  are 
applied,  it  is  evident  that  it  is  neither  a  difference  in  the  pro- 
cess of  forming,  nor  the  essential  constitution  of  concepts,  but 
a  much  more  external  circumstance  which  is  expressed  by  the 
terms.  Even  Mill's  proposition  to  restore  scholastic  usage  of 
them  may  be  opposed  to  the  practical  consideration  that,  in 
the  modified  sense  in  which  they  refer  to  the  degree  of  apply- 
ing the  process  of  abstraction,  they  have  already  obtained, 
through  common  usage,  a  right  to  a  place  in  the  language  of 
science,  as  well  as  in  practical  life,  which  at  the  same  time  in- 
dicates a  demand  for  a  corresponding  logical  distinction.  If 
we  regard  only  the  present  practice  of  language,  it  will  not 
appear  doubtful  that  we  have  here  to  do  chiefly  with  the  rela- 
tion of  a  concepition  to  its  representative  percept.  So  long  as 
the  latter  consists  in  a  sensible  presentation  in  which  the  essen- 
tial elements  of  the  concept  are  realized,  and  not  merely  in 
the  word  denoting  it,  we  name  it  concrete.  But,  on  the  other 
hand,  so  soon  as  the  written  or  spoken  word  becomes  a  single 
sign  or  symbol  for  the  concept,  it  is  abstract.  In  other  words, 
those  concepts  are  abstract  to  which  no  adequate  representa- 
tive percept  corresponds,  and  for  which,  in  thought,  we  can 
only  choose  an  external  and  apparently  arbitrary  symbol.  In 
this  sense  we  should  doubtlessly  describe  such  a  concept  as 
'  man '  or  '  animal '  as  concrete,  and  such  a  concept  as 
'  humanity '  as  abstract.  But  in  opposition  to  scholastic 
usage  we  could  call  '  the  righteous'  as  well  as  'righteous- 
ness' by  the  name  of  abstract.  And  further,  an  individual 
concept  would  most  frequently  be  concrete  at  the  same  time, 
while  a  concrete  term  would  very  frequently  be  general.  Also 
it  would  certainly  happen  that,  in  individual  cases,  the  distinc- 
tion would  remain  indeterminate  or  doubtful.     Thoimh  words 


42  ELEMENTS  OF  LOGIC1 

have  gradually  developed  into  signs  of  abstract  conceptions, 
and  though,  as  the  history  of  the  change  of  meaning  every- 
where shows,  abstract  conceptions  have  been  developed  from 
concrete,  why  should  we  not  at  times  meet  a  conception  which 
remains  in  the  intermediate  stage  of  development  ?  Concep- 
tions like  'machine,'  'weight,'  etc.,  may,  in  fact,  be  com- 
pletely abstract  to  one  person,  and  in  another  be  attached  to  a 
sense  picture,  and  consequently  concrete.  From  this  a  single 
conclusion  is  evident ;  namely,  that  this  distinction  has  little 
importance  for  the  nature  of  a  conception,  but  that  it  has  just 
as  great  an  importance  for  the  development  of  abstract  no- 
tions. This  development  depends  upon  the  constant  ap- 
plication of  the  same  process  which  we  find  is  active  in  the 
origin  of  all  conceptions,  even  the  most  individual  and  con- 
crete." * 

A  pure  abstract  concept,  therefore,  I  shall  consider  as  de- 
fined to  be  a  quality  or  attribute  conceived  and  treated  as  a 
substantive  or  thing.  A  mixed  abstract  and  concrete  term 
will  then  be  capable  of  either  reference  according  to  circum- 
stances and  the  degree  of  its  definiteness.  It  is  general  terms 
that  are  so  frequently  of  this  mixed  character  and  that  are  the 
source  of  confusion  in  Logic.  The  pure  forms  also  give  rise  to 
a  certain  order  of  errors.  In  the  first  place,  there  is  the  danger 
of  treating  abstract  conceptions  as  if  they  represented  inde- 
pendent realities.  Thus  such  ideas  as  "truth,"  "beauty," 
"excellence,"  "virtue,"  "nature,"  "law,"  etc.,  are  often  spoken 
of  by  writers  as  if  they  represented  existences  independent 
of  particular  objects  or  persons,  of  which  they  are  in  reality 
only  qualities.  The  error  here  is  first  in  the  conception,  and 
reasoning  is  subsequently  affected  by  it  as  the  conclusion  is 
always  vitiated  or  validated  by  the  character  of  the  premises. 
But  modern  thought  is  better  provided  against  the  confusion 
of  the  abstract  and  the  concrete  than  ancient  and  mediaeval 
speculation,  when  it  concerns  the  pure  concrete  and  the  pure 
abstract  concepts.  This  is  not  the  case  with  the  second  form  of 
error  which  arises  from  the  confusion  of  the  concrete  and  the 
*  Wundt:  Logik,  Bd.  I.,  Zweiter  Abschnitt,  Cap.  I.,  §  3,  p.  97. 


TERMS  OR  CONCEPTS,   AND  THEIR  KINDS         43 

abstract  in  the  same  terms  and  propositions.  One  person  may 
have  the  abstract  and  another  the  concrete  aspect  in  conscious- 
ness when  using  the  concept,  and  unless  what  is  affirmed  of  it 
be  true  of  both  aspects  there  will  be  a  difference  of  opinion 
between  the  two,  or  one  will  be  guilty  of  error  in  thought  and 
reasoning.  This  is  less  apparent  in  conceptions  than  in  prop- 
ositions. Hence  it  is  more  frequent  that  abstract  thoughts, 
propositions,  or  principles  should  produce  error  than  the  mere 
concept  alone,  or  the  confusion  of  the  abstract  and  concrete 
aspects  of  it,  although  the  error  in  the  use  of  principles  must 
begin  with  an  error  in  the  use  of  the  concept.  But  as  pos- 
sible sources  of  fallacy  in  the  conception  of  mixed  terms 
we  have  such  general  concepts  as  "religion,"  "institution," 
"home,"  "history,"  "organism,"  "socialism,"  where  we  may 
not  be  assured  whether  it  is  the  concrete  or  the  abstract  form 
which  is  prominent  in  consciousness.  But  all  this  is  brought 
out  more  clearly  in  propositions  than  in  concepts.  Thus  if  I 
take  the  proposition,  "Religion  is  useful,"  I  may  mean  "relig- 
ion "  in  the  abstract  ideal  form,  or  I  may  mean  in  its  particular 
concrete  form ;  that  is,  all  religions  and  denominations.  Or  I 
may  say,  "Governments  are  good  institutions,"  and  mean  either 
all  particular  governments,  or  government  in  the  abstract.  In 
the  concrete,  governments  might  be  very  bad,  while  in  the  ab- 
stract they  might  be  very  good.  "  Government,"  as  an  abstract 
concept,  is  only  an  ideal  quality  of  the  aggregate  of  men  or  of 
those  in  power,  and  so,  in  the  concrete,  can  never  be  better 
than  the  men  composing  it.  But  in  speaking  and  writing  of  it, 
it  is  often  convenient  to  treat  the  concept  independently  of  its 
reference  to  the  men  who  made  it,  and  then  it  is  considered 
in  the  abstract.  As  long  as  the  men  who  compose  government 
are  bad,  I  can  say  that  "  government  is  a  good  institution  " 
only  when  I  consider  it  abstractly.  Concretely  government 
could  only  be  what  the  men  are  who  compose  it.  The  same 
observations  apply  to  any  other  mixed  concepts  and  proposi- 
tions. This  will  also  be  true,  but  perhaps  in  a  less  degree,  of 
such  terms  as  "man,"  "animal,"  "vegetable,"  etc.,  for  we 
occasionally  give  them  an  abstract  import,  although  their  con- 


44  ELEMENTS  OF  L0G10 

crete  reference  may  be  the  most  frequent.  But  the  signifi- 
cance and  importance  of  these  distinctions  will  be  more  a\> 
parent  when  we  come  to  consider  the  essence  and  accidents  of 
concepts.  For  the  difference  between  the  abstract  and  the 
concrete  is  closely  related  or  connected  with  that  between  es- 
sence and  accident. 

4th.  Positive,  Negative,  Privative,  and  Nego-positive 
Terms. — The  usual  division  of  terms  in  this  respect  is  into 
Positive,  Negative,  and  Privative,  but  I  add  the  fourth  for 
reasons  which  will  appear  in  the  sequel.  I  shall  first  define 
and  illustrate  them. 

Positive  terms  or  concepts  are  those  which  signify  the  exist- 
ence, presence,  or  possession  of  certain  qualities  ;  for  example, 
"good,"  "pure,"  "excellence,"  "metal,"  "organic,"  "human," 
etc.  A  positive  term  is  thus  one  which  is  so  both  grammati- 
cally and  logically,  or  both  in  form  and  matter. 

Negative  terms  or  concepts  are  those  which  signify  the 
absence  of  certain  qualities,  as  "  impure,"  "  inorganic,"  "  un- 
human,"  "  non-metal,"  "  ingratitude,"  "  insipid,"  etc.  They  are 
thus  negative  in  both  form  and  matter. 

Privative  terms  or  concepts  are  those  which  denote  the  de- 
privation of  certain  qualities  once  possessed  or  the  normal 
characteristic  of  the  subject.  Thus  "  deaf,"  "  dumb,"  "  blind," 
"  dead,"  are  privative  terms.  In  so  far  as  they  denote  the 
absence  of  qualities  they  are  negative  terms.  But  they  differ 
in  their  form  from  negative  terms,  although  materially  con- 
sidered they  are  only  modifications  of  them.  They  ma}r  be 
more  strictly  defined  as  terms  which  are  positive  in  their 
form,  but  negative  in  their  matter.  This  will  be  apparent 
from  the  illustrations. 

Nego-positive  terms  or  concepts  are  those  which  denote  the 
presence  of  a  positive  quality,  although  they  appear  to  be  neg- 
ative. Thus  "  disagreeable,"  "  inconvenience,"  "  infamous," 
"ignorant,"  "displeasure,"  "immediate,"  "undone,"  etc.  They 
are  thus  negative  in  form  and  positive  in  matter.  They  can  in 
many  cases  be  most  easily  distinguished  by  comparing  them 
with   their   corresponding  positive  conceptions.     Thus  "un- 


TERMS  OR   CONCEPTS,   AND   THEIR  KINDS        45 

happiness"  and  "invaluable"  have  their  equivalents  in  "mis- 
ery" and  "costly,"  both  of  which  are  j^ositive. 

Some  terms  may  be  taken  in  either  a  negative  or  a  nego-posi- 
tive  sense.  Thus  "uncertain,"  "unhealthy,"  "unpleasant," 
"  indistinct,"  may  be  conceived  as  the  negatives  of  "  certain," 
"healthy,"  "pleasant,"  "distinct,"  or  as  the  nego-positive 
equivalents  of  "  doubtful,"  "  sickly,"  "  painful,"  "  obscure." 
They  are,  however,  the  same  modification  of  positive  concep- 
tions that  privative  terms  are  of  negatives.  In  fact,  privative 
and  nego-positive  terms  are  simply  mixed  concepts,  having  an 
element  from  each  of  the  other  two,  but  combining  them  in  a 
reversed  relation.  In  the  broader  sense,  therefore,  we  can 
divide  all  terms  into  positive  and  negative.  In  this  broader 
sense  the  symbols  or  indices  of  negative  terms  are  in,  un, 
non,  less,  dis,  a  or  an,  anti,  and  sometimes  de,  and  perhaps 
mis. 

The  following  tabular  outlines  will  give  the  divisions  and 
indicate  their  nature  and  relations  to  each  other. 


C  Positive   =  Positive  in  both  form  and  matter. 
Terms        )  Negative  =  Negative  in  both  form  and  matter. 
or  Concepts.  ]  Privative  =  Positive  in  form  and  negative  in  matter. 

I,  Nego-positive  =  Negative  in  form  and  positive  in  matter. 

f  Pure  i  Positive-  Tprms  f  j  Pure  =  Simple  positive. 

Terms        rure      Negative.  Ierms    Positive    1  ,  -p  .     .. 

or        -  Privative.  nor     \  Mixed    £nvatlve:. . 

Concepts    Mixed  \     Nego-  f^  1  Negative  J  '  <  Nego-positive. 

[  (  positive.  I  ( Pure  =  Simple  Negative 


The  reason  for  distinguishing  a  separate  class  of  nego- 
positive  tei'ms  is  their  liability  at  times  to  be  mistaken  for 
negatives,  and  the  confusion  often  incident  to  the  transition, 
or  immediate  inference  from  a  purely  negative  conception  to 
one  which  is  really  positive  in  its  meaning,  although  expressed 
by  the  same  term.  Thus  if  we  were  to  argue  that  a  thing  or  an 
act  is  bad  because  it  is  not-good,  we  should  be  committing 
a  fallacy,  considering  that  "not-good"  is  a  purely  negative 
conception.  The  same  remark  applies  to  the  passage  from 
the  not-just  to  the  unjust,  and  from  the  non-moral  or  not-moral 


46  ELEMENTS  OE  LOOIO 

to  the  immoral.  For  instance,  physical  acts  are  "  not-moral," 
but  they  are  also  not  immoral.  In  ethics  there  are  three 
classes  of  acts — the  moral,  the  non-moral,  and  the  immoral — or 
in  stoical  parlance — good,  indifferent,  and  bad.  Or  we  might 
divide  them  first  into  moral  and  not-moral,  and  subdivide  the 
latter  into  non-moral  and  immoral.  Similarly  we  may  recog- 
nize just,  not-just,  and  unjust  acts.  The  distinction  is  here 
the  same.  Also  in  many  other  departments  of  thought  this 
triple  division  is  the  proper  one,  although  the  usual  division  is 
dichotonious  instead  of  trichotomous.  Where  it  occurs  the 
negative  and  nego-positive  conceptions  do  not  necessarily 
coincide.  It  is  true,  however,  that  a  negative  term  is  often 
used  as  the  equivalent  of  the  nego-positive.  Thus  we  some- 
times describe  an  act  as  "  not-just "  when  we  think  of  it  as 
"  unjust."  This  is  when  we  do  not  think  of  the  conception  as 
infinitated.  By  an  infinitated  conception  I  mean  one  which 
comprehends  all  other  possible  objects  in  the  world  than  those 
denoted  by  the  contradictory  positive  term.  Thus  in  its 
strict  meaning  and  extension  the  negative  "not-just"  will  in- 
clude all  other  things  in  the  universe  than  those  expressed  by 
"  just."  Hence  among  "  not-just  "  might  be  found  material 
objects,  such  as  trees,  stones,  houses,  etc.  Again,  "  not-house  " 
would  include  all  other  things  in  the  universe  than  the  term 
"  house  ; "  and  so  on  with  all  similar  conceptions.  The  infini- 
tated concept  is  simply  all  else  than  what  is  expressed  by  the 
positive.  But  often  a  negative  concept  is  used  either  as  a 
euphemism,  or  as  an  equivalent  of  the  nego-positive.  In  such 
cases  they  are  clearly  convertible.  Thus  wherever  "  unpleas- 
ant "  is  thought  of  as  "  painful,"  the  two  concepts  can  be  sub- 
stituted for  each  other.  Illustrations  of  this  will  appear  in 
Conversion.  But  when  the  negative  term  is  infinitated  this 
substitution  cannot  be  made,  and  it  is  the  business  of  the  stu- 
dent and  the  logician  to  be  on  the  alert  for  the  confusion  inci- 
dent to  such  a  procedure. 

The  terms  "  greater,"  "  less,"  and  "  equal "  deserve  a  brief 
notice.  They  are  all  positive  concepts,  but  taken  in  rela- 
tion to  each  other,  as  they  must  be,  they  are  relative  terms, 


TERMS  OR  CONCEPTS,   AND   THEIR  KINDS         47 

and  must  be  considered  in  the  next  section.  But  being  rela- 
tive concepts  we  may  say  that  any  two  of  them  are  the 
negatives  of  the  third.  But  their  pure  negatives  are  the 
intinitated  forms  of  their  positives,  and  it  is  only  as  rela- 
tives that  any  of  them  can  be  considered  the  negative  of  the 
others. 

What  has  been  said  of  the  terms  "  greater,"  "  less,"  and 
"equal"  can  be  said  of  a  large  number  of  terms  in  the  lan- 
guage. Indeed,  every  term  may  be  said  to  be  negative  in  rela- 
tion to  all  other  terms,  except  its  own  equivalents  or  synonyms. 
It  will  not  need  to  be  so  in  its  form,  but  only  in  its  meaning  as 
related  to  those  terms  which  do  not  express  the  same  concej)t. 
Hence,  while  we  should  call  "  horse  "  a  positive  term,  in  relation 
to  man  it  would  be  negative,  because  it  would  be  negatively 
conceived  as  "not  man  ;"  "tree,"  as  not  a  lion  ;  "  external  in- 
fluences "  as  "  not  internal  influences,"  etc.  Very  often,  from 
the  mere  habit  of  conceiving  different  concepts  as  excluding 
each  other,  we  assume  that  particular  cases  represent  contrary 
concepts,  one  the  negative  or  contradictory  of  each  other, 
when  in  reality  they  are  not  so.  Many  errors  of  opinion  and 
of  reasoning  are  incident  to  this  mistake.  We  require  always 
to  examine  how  far  the  meaning  of  concepts  excludes  that  of 
other  terms,  and  not  to  assume  a  negative  relation  from  the 
mere  fact  that  it  generally  so  exists.  A  very  large  number  of 
concepts  have  this  relation  ;  but  the  form  of  the  term  will  be 
no  adequate  criterion  of  the  fact,  and  we  must  examine  the 
matter  of  thought  in  given  cases,  in  order  to  decide  the  ques- 
tion for  practical  instances. 

5th.  Absolute  and  Relative  Terms  or  Concepts. — 
This  distinction,  as  usually  applied,  does  not  have  much  im- 
poi'tance  in  Logic.  Hence  it  may  be  dismissed  very  briefly. 
There  is,  perhaps,  a  more  comprehensive  sense  in  which  the 
distinction  is  a  valuable  one.  But  it  does  not  apply  in  any 
special  way  to  the  few  terms  known  particularly  as  relative. 
Most  terms  are  absolute  concepts  in  the  ordinary  sense  of  the 
word. 

"  Absolute  "  means  literally  that  which  is  severed  from  all 


48 


ELEMENTS  OF  LOGIO 


dependence  on  another.  Hence  an  absolute  term  is  one  which 
expresses  what  can  be  thought  of  by  itself,  and  which  does  not 
have  its  meaning  determined  by  comparison  with  some  corre- 
late object  of  consciousness.  Thus  "  man,"  "  tree,"  "  earth," 
"star,"  "book,"  etc.,  are  absolute  concepts.  They  have  no 
correlatives  implied  in  them.  They  may  denote  things  which 
exist,  both  in  thought  and  reality,  in  relation  to  something 
else.  But  this  related  object  is  merely  associated  with  the 
conception  and  may  not  be  necessary  to  the  meaning  of  the 
term,  or  to  its  presentation  in  consciousness.  Quite  the  con- 
trary is  true  of  relative  terms. 

A  "  relative  "  term  or  concept  is  one  which  denotes  an  object 
that  cannot  be  thought  of  without  reference  to  some  other 
object,  which  is  its  correlate.  Thus  "father"  and  "son," 
"parent"  and  "  child  "  are  correlatives.  Each  has  its  meaning 
determined  in  relation  to  the  other.  Again,  the  terms  "  mon- 
arch," "shepherd,"  "master,"  "teacher,"  "subject,"  "slave," 
"  eldest,"  etc.,  are  relative.  Their  correlates  can  easily  be  re- 
marked. It  will  be  apparent  also  how  "greater"  and  "less" 
may  be  regarded  as  relative.  But  as  little  confusion  is  incident 
to  logical  processes  connected  with  the  use  of  relative  terms 
as  here  denned,  the  subject  does  not  require  further  consider- 
ation. 

In  a  broader  sense,  as  I  have  remarked,  every  term  is  relative. 
It  is  first  relative  to  its  negative  concept,  and  we  often  find  this 
means  a  convenient  one  for  defining  a  term.  But  it  is  never 
completely  satisfactory,  because  a  true  definition  demands  a 
statement  of  the  positive  content  of  a  concept.  The  negative 
term,  however,  is  a  very  convenient  one  for  representing  the 
actual  relation  sustained  by  every  concept.  Further  than  this, 
every  object  may  be  said  to  exist  in  a  relation  to  some  other, 
or  all  other  objects,  and  the  relation  between  two  or  more  ob- 
jects may  be  so  close  as  to  mutually  affect  each  other's  meaning. 
Thus  "  day"  and  "night,"  "joy"  and  "sorrow,"  "pleasure" 
and  "pain,"  "  true  "  and  "false,"  are  conceived  as  relatives  by 
contrast.  But  while  they  may  be  so  related  in  thought,  the 
material  or  real  existence  of  one  of  the  correlates  is  not  im- 


TERMS  OR   CONCEPTS,   AND   THEIR  KINDS  49 

plied  by  the  other.  Such  relation  as  may  exist  between  other 
classes  of  conceptions  is  too  remote  and  unimportant  for  logic 
to  take  any  special  notice  of  it.* 

*  For  discussion  of  technical  problems  connected  with  the  qualities  of 
terms  the  student  may  consult  the  following  works:  Mill:  Logic,  Bk.  I., 
Chap.  II.  ;  Venn  :  Empirical  Logic,  Chap.  VII.  ;  Keynes :  Formal  Logic, 
Part  I.,  Chaps.  I. -III.,  pp.  7-50;  Jevons :  Principles  of  Science,  Bk.  I., 
Chap.  II.;  Wundt:  Logik,  Zweiter  Abschnitt,  Chaps.  I.-IV.,  pp.  86-134. 
4 


CHAPTER  IV. 

THE   AMBIGUITY  OF   TERMS 

In  this  chapter  I  shall  do  little  more  than  transcribe  the 
language  of  Jevons  upon  the  subject.  He  has  said  about  all 
that  a  practical  Logic  requires  to  have  said  upon  it.  The  im- 
portance of  considering  it  ought  to  be  apparent  to  everyone, 
and  will  be  so  to  those  who  know  or  suspect  that  the  majority 
of  fallacies  in  our  reasoning  turn  upon  the  ambiguous  use  of 
words.  Most  of  our  controversies  are  logomachies  on  the  same 
account.  We  think  we  are  employing  the  same  conceptions 
when  we  are  only  using  the  same  terms  ;  the  difference  be- 
tween our  conceptions,  in  sprite  of  the  identity  in  language, 
may  be  as  great  as  between  different  words.  Different  terms 
also  may  be  used  for  the  same  concepts,  and  thus  give  rise  to  a 
converse  error.  But  the  most  frequent  source  of  error  is  am- 
biguity.    A  syllogism  will  illustrate  it : 

No  designing  person  ought  to  be  trusted. 
Engravers  are  by  profession  designers. 
.'.  They  ought  not  to  be  trusted. 

It  is  easy  enough  to  detect  in  such  cases  the  source  of  the 
fallacy.  But  there  are  instances  where  the  ambiguity  is  more 
subtle,  and  requires  a  keener  logical  insight  for  its  detection. 
In  profounder  subjects  of  speculation  it  requires  a  wide  knowl- 
edge of  the  use  of  language  and  a  thorough  acquaintance  with 
the  laws  and  processes  of  the  mind.  In  fact,  a  reasoner  should 
always  be  on  the  alert  for  the  ambiguous  use  of  terms,  and  he 
cannot  have  the  possibility  of  such  a  source  of  confusion  better 
indicated  than  by  a  few  observations  upon  the  simj)lest  words 
of  the  language. 


THE  AMBIGUITY  OF  TERMS  51 

We  may  divide  terms  into  unifocal  and  equivocal.  A  uni- 
vocal  term  is  one  with  but  a  single  meaning,  which  is  exposed 
to  no  mistake  of  interpretation.  An  equivocal  term  is  one  with 
more  than  a  single  meaning.  Very  little  observation  is  neces- 
sary to  show  that  very  few  terms  come  under  the  class  univo- 
cal.  Proper  names,  and  therefore  singular  terms,  are  almost 
the  only  conceptions  with  an  unmistakable  import,  and  even 
some  of  these  are  equivocal.  A  general  concept  applying  only 
to  a  single  class  of  individuals  of  exactly  the  same  kind  may 
be  univocal.  Thus  President  Lincoln,  St.  Paul's  Cathedral, 
Berlin,  are  univocal.  But  Washington  may  be  equivocal  as 
applying  to  a  person,  a  city,  or  a  state.  Of  common  terms 
that  are  univocal,  Jevons  thinks  that  they  are  chiefly  found  in 
technical  and  scientific  language.  He  enumerates  "  steam- 
engine,"  "gasometer,"  and  "  railway  train,"  and  in  common 
life  such  terms  as  "  penny,"  "  mantelpiece,"  "  tea-cup,"  "  bread," 
and  "  butter."  "  Cathedral "  is  probably  univocal,  or  of  one 
logical  meaning  only.  But  "  church  "  is  equivocal,  as  referring 
sometimes  to  a  building,  and  sometimes  to  a  corporate  body  of 
men.  Compared  with  the  equivocal  terms,  however,  the  uni- 
vocal are  very  few. 

From  this  point  we  may  simply  quote  Jevons.  He  begins 
with  a  division  of  equivocal  terms.  "  We  may  distinguish," 
he  says,  "  three  classes  of  equivocal  words,  according  as  they 
are — 

"  1.  Equivocal  in  sound  only. 

"  2.  Equivocal  in  spelling  only. 

"  3.  Equivocal  both  in  sound  and  spelling. 

"The  first  two  classes  are,  comparatively  speaking,  of  very 
slight  importance,  and  do  not  often  give  rise  to  serious  error. 
They  produce  what  we  call  trivial  mistakes.  Thus  we  may 
confuse,  when  spoken  only,  the  words  right,  wright,  and  rite 
(ceremony)  ;  also  the  words  rein,  rain,  and  reign  ;  might,  mite, 
etc.  Owing  partly  to  defects  of  pronunciation,  mistakes  are 
not  unknown  between  the  four  words  air,  hair,  hare,  and  heir. 

"  Words  equivocal  in  spelling,  but  not  in  sound,  are  such  as 
tear  (drop),   and  tear,  pronounced  tare,  meaning  a  rent  in 


52  ELEMENTS  OF  LOOIO 

cloth  ;  or  lead,  the  metal,  and  lead,  as  in  following  the  lead  of 
another  person.  As  little  more  than  momentary  misapprehen- 
sion, however,  can  arise  from  such  resemblance  of  words,  we 
shall  pass  at  once  to  the  class  of  words  equivocal  both  in  sound 
and  spelling.  These  I  shall  separate  into  three  groups,  accord- 
ing as  the  equivocation  arises. 

"  (a)  From  the  accidental  confusion  of  different  words. 

"  (b)  From  the  transfer  of  meaning  by  the  association  of 
ideas. 

"  (c)  From  the  logical  transfer  of  meaning  to  analogous  ob- 
jects. 

"  (a)  Under  the  first  class  we  place  a  certain  number  of  curi- 
ous but  hardly  important  cases  in  which  ambiguity  has  arisen 
from  the  confusion  of  entirely  different  words,  derived  from 
different  languages  or  from  different  roots  of  the  same  lan- 
guage, but  which  have  in  the  course  of  time  assumed  the 
same  sound  and  spelling.  Thus  the  word  mean  denotes  either 
that  which  is  medium  or  mediocre,  from  the  French  moyen 
and  the  Latin  medius,  connected  with  the  Anglo-Saxon  mid  or 
middle  ;  or  it  denotes  what  is  low-minded  and  base,  being  then 
derived  from  the  Anglo-Saxon  gemoene,  which  means  'that 
belonging  to  the  moene  or  many,'  whatever,  in  short,  is  vulgar. 
The  verb  to  mean  can  hardly  be  confused  with  the  adjective 
mean,  but  it  comes  from  a  third  distinct  root,  probably  con- 
nected with  the  Sanscrit  verb  to  think. 

"  As  other  instances  of  this  casual  ambiguity  I  may  mention 
rent,  a  money  payment,  from  the  French  rente  (rendre,  to  re- 
turn), or  a  tear,  the  result  of  the  action  of  rending,  this  word 
being  of  Auglo-Saxon  origin  and  one  of  the  numerous  class 
beginning  in  r  or  wr,  which  imitate  more  or  less  perfectly  the 
sound  of  the  action  which  they  denote.  Pound,  from  the 
Latin  pondus,  a  weight,  is  confused  with  pound,  in  the  sense  of 
a  village  pinfold  for  cattle,  derived  from  the  Saxon  pydan,  to 
pen  up.  Fell,  a  mountain,  is  a  perfectly  distinct  word  from 
fell,  a  skin  or  hide  ;  and  £w tee,  a  throb  or  beating,  and  pulse, 
peas,  beans,  or  potage,  though  both  derived  from  the  Greek 
or  Latin,  are  probably  quite  unconnected  words.     It  is  curi- 


THE  AMBIGUITY   OF  TERMS  53 

ous  that  gin,  in  the  meaning  of  trap  or  machine,  is  a  con- 
tracted form  of  engine,  and  when  denoting  the  spirituous 
liquor  is  a  corruption  of  Geneva,  the  place  where  the  spirit 
was  first  made. 

"  Certain  important  cases  of  confusion  have  been  detected  in 
grammar,  as  between  the  numeral  one,  derived  from  an  Aryan 
root,  through  the  Latin  unus,  and  the  indeterminate  pronoun 
one  (as  in  '  one  ought  to  do  one's  duty '),  which  is  really  a  cor- 
rupt form  of  the  French  word  homme  or  man.  The  Germans 
to  the  present  day  use  man  in  this  sense,  as  in  'man  sagt,'  i.e., 
one  says. 

"  (b)  By  far  the  largest  part  of  equivocal  words  have  become 
so  by  a  transfer  of  the  meaning  from  the  thing  originally  de- 
noted by  the  word  to  some  other  thing  habitually  connected 
with  it  so  as  to  become  closely  associated  in  thought.  Thus 
in  Parliamentary  language  the  House  means  either  the  cham- 
ber in  which  the  members  meet,  or  it  means  the  body  of  mem- 
bers who  happen  to  be  assembled  in  it  at  any  time.  Similarly 
the  word  church  originally  denoted  the  building  («uptaKoV, 
the  Lord's  House)  in  which  religious  worshippers  assemble, 
but  it  has  thence  derived  a  variety  of  meanings  ;  it  may  mean 
a  particular  body  of  worshippers  accustomed  to  assemble  in 
any  one  place,  in  which  sense  it  is  used  in  Acts  xiv.  23  ;  or 
it  means  any  body  of  persons  holding  the  same  religious 
opinions,  and  connected  in  one  organization,  as  in  the  Angli- 
can, or  Greek,  or  Roman  Catholic  Church  ;  it  is  also  some- 
times used  so  as  to  include  the  laity  as  well  as  the  clergy  ;  but 
more  generally  perhaps  the  clergy  and  religious  authori- 
ties of  any  sect  or  country  are  so  strongly  associated  with  the 
act  of  worship  as  to  often  be  called  the  church  par  excellence. 
It  is  quite  evident,  moreover,  that  the  word  entirely  differs  in 
meaning  according  as  it  is  used  by  a  member  of  the  Anglican, 
Greek,  Roman  Catholic,  Scotch  Presbyterian,  or  any  other  ex- 
isting church. 

"  The  word  foot  has  suffered  several  curious  but  very  evi- 
dent transfers  of  meaning.  Originally  it  denoted  the  foot  of 
a  man  or  an  animal,  and  is  probably  connected  in  a  remote 


54  ELEMENTS  OF  LOGIC 

manner  with  the  Latin  jpes,  pedis,  and  the  Greek  irovs,  7roSos ; 
but  since  the  length  of  the  foot  is  naturally  employed  as  a 
rude  measure  of  length  it  came  to  be  applied  to  a 
fixed  measure  of  length ;  and  as  the  foot  is  at  the  bottom  of 
the  body  the  name  was  extended  by  analogy  to  the  foot  of  a 
mountain,  or  the  feet  of  a  table  ;  by  a  further  extension,  any 
position,  plan,  reason,  or  argument  on  which  we  place  our- 
selves and  rely,  is  called  the  foot  or  footing.  The  same  word 
also  denotes  soldiers  who  fight  upon  their  feet,  or  infantry, 
and  the  measured  part  of  a  verse  having  a  definite  length. 
That  these  very  different  meanings  are  naturally  connected 
with  the  original  meaning  is  evident  from  the  fact  that  the 
Latin  and  Greek  words  for  foot  are  subject  to  exactly  similar 
series  of  ambiguities. 

"It  would  be  a  long  task  to  trace  out  completely  the  various 
and  often  contradictory  meanings  of  the  word  fellow.  Origi- 
nally a  fellow  was  what  follows  another,  that  is,  a  companion  ; 
thus  it  came  to  mean  the  other  of  a  pair,  as  one  shoe  is  the 
fellow  of  the  other,  or  simply  an  equal,  as  when  we  say  that 
Shakespeare  'hath  not  a  fellow.'  From  the  simple  meaning 
of  companion,  again,  it  comes  to  denote  vaguely  a  person,  as  in 
the  question,  '  What  fellow  is  that  ?  '  but  then  there  is  a  curious 
confusion  of  depreciatory  and  endearing  power  in  the  word  ; 
when  a  man  is  called  a  mere  felloiv,  or  simply  afelloiv  in  a  par- 
ticular tone  of  voice,  the  name  is  one  of  severe  contempt ;  alter 
the  tone  of  voice  of  the  connected  words  in  the  least  degree, 
and  it  becomes  one  of  the  most  sweet  and  endearing  appella- 
tions, as  when  we  speak  of  a  dear  or  good  fellow.  "We  may 
still  add  the  technical  meanings  of  the  name  as  applied  in  the 
case  of  a  Fellow  of  a  College  or  of  a  learned  society. 

"  Another  good  instance  of  the  growth  of  a  number  of  differ- 
ent meanings  from  a  single  root  is  found  in  the  word  post. 
Originally  a  post  was  something  2^osited,  or  placed  firmly  in 
the  ground,  such  as  an  upright  piece  of  wood  or  stone  ;  such 
meaning  still  remains  in  the  cases  of  a  lamp-post,  a  gate-post, 
signal-post,  etc.  As  a  post  would  often  be  used  to  mark  a 
fixed  spot  of  ground,  as  in  a  mile-post,  it  came  to  mean  the 


THE  AMBIGUITY  OF  TERMS  55 

fixed  or  appointed  place  where  the  post  was  placed,  as  in  a 
military  post,  the  post  of  danger,  or  honor,  etc.  The  fixed 
places  where  horses  were  kept  in  readiness  to  facilitate  rapid 
travelling  during  the  times  of  the  Roman  empire  were  thus 
called  posts,  and  thence  the  whole  system  of  arrangement  for 
the  conveyance  of  persons  or  news  came  to  be  called  the  posts. 
The  name  has  retained  an  exactly  similar  meaning  to  the  pres- 
ent day  in  most  parts  of  Europe,  and  we  still  use  it  in  post- 
chaise,  post-boy,  post-horse,  and  postilion.  A  system  of  post 
conveyance  for  letters  having  been  organized  for  about  two 
centuries  in  England  and  other  countries,  this  is  perhaps  the 
meaning  most  closely  associated  with  the  word  post  at  present, 
and  a  number  of  expressions  have  arisen,  such  as  post-office, 
postage,  postal-guide,  postman,  postmaster,  postal-telegraph, 
etc.  Curiously  enough,  we  now  have  iron  letter-posts,  in 
which  the  word  post  is  restored  exactly  to  its  original  mean- 
ing. 

"  Although  the  words  described  above  were  selected  on  ac- 
count of  the  curious  variety  of  their  meanings,  I  do  not  hesi- 
tate to  assert  that  the  majority  of  common  nouns  possess  vari- 
ous meanings  in  greater  or  less  number.  Dr.  Watts,  in  his 
'  Logic,'  suggests  that  the  words  book,  Bible,  fish,  house,  and 
elephant  are  univocal  terms,  but  the  reader  would  easily 
detect  ambiguities  in  each  of  them.  Thus  fish  bears  a  very 
different  meaning  in  natural  history  from  what  it  does  in  the 
mouth  of  unscientific  persons,  who  include  under  it  not  only 
true  fishes,  but  shell-fish  or  mollusca,  and  the  cetacea,  such  as 
whales  and  seals,  in  short,  all  swimming  animals,  whether  they 
have  the  character  of  true  fish  or  not.  Elephant,  in  a  station- 
er's or  bookseller's  shop,  means  a  large  kind  of  paper  instead 
of  a  large  animal.  Bible  sometimes  means  any  particular  copy 
of  the  Bible,  sometimes  the  collection  of  works  constituting 
the  Holy  Scriptures.  The  word  man  is  singularly  ambiguous  ; 
sometimes  it  denotes  man  as  distinguished  from  woman  ;  at 
other  times  it  is  certainly  used  to  include  both  sexes  ;  and  in 
certain  recent  election  cases  lawyers  were  unable  to  decide 
whether  the  word  man,  as  used  in  the  Reform  Act  of  1867, 


56  ELEMENTS  OF  LOGIC 

ought  or  ought  not  to  be  interpreted  so  as  to  include  women. 
On  other  occasions  man  is  used  to  denote  an  adult  male  as 
distinguished  from  a  boy,  and  it  also  often  denotes  one  who 
is  emphatically  a  man  as  possessing  a  masculine  character 
(heroic).  Occasionally  it  is  used  in  the  same  way  as  groom, 
for  a  servant,  as  in  the  proverb,  "Like  master,  like  man."  At 
other  times  it  stands  specially  for  husband. 

"  (c)  Among,  ambiguous  words  we  must  thirdly  distinguish 
those  which  derive  their  various  meanings  in  a  somewhat  dif- 
ferent manner,  namely,  by  analogy  or  real  resemblance.  When 
we  speak  of  a  sweet  taste,  a  sweet  flower,  a  sweet  tune,  a  sweet 
landscape,  a  sweet  face,  a  sweet  poem,  it  is  evident  that  we 
apply  one  and  the  same  word  to  very  different  things  ;  such  a 
concrete  thing  as  lump-sugar  can  hardly  be  compared  directly 
with  such  an  intellectual  existence  as  Tennyson's  May  Queen. 
Nevertheless,  if  the  word  sweet  is  to  be  considered  ambiguous, 
it  is  in  a  different  way  from  those  we  have  before  considered, 
because  all  the  things  are  called  sweet  on  account  of  a  peculiar 
pleasure  which  they  yield,  which  cannot  be  described  otherwise 
than  by  comparison  with  sugar.  In  a  similar  way  we  describe 
a  pain  as  sharp,  a  disappointment  as  bitter,  a  person's  temper 
as  sour,  the  future  as  bright  or  gloomy,  an  achievement  as 
brilliant ;  all  these  adjectives  implying  comparison  with  bodily 
sensations  of  the  simplest  kind.  The  adjective  brilliant  is  de- 
rived from  the  French  briller,  to  glitter  or  sparkle  ;  and  this 
meaning  it  fully  retains  when  we  speak  of  a  brilliant  diamond, 
a  brilliant  star,  etc.  But  by  what  subtle  analogy  is  it  that  we 
speak  of  a  brilliant  position,  a  brilliant  achievement,  brilliant 
talents,  brilliant  style  !  We  cannot  speak  of  a  clear  explana- 
tion, indefatigable  perseverance,  perspicuous  style,  or  sore 
calamity,  without  employing  in  each  of  these  expressions  a 
double  analogy  to  physical  impressions,  actions,  or  events." 

Continuing  the  discussion  in  a  later  chapter  on  "  The  Growth 
of  Language,"  the  same  author  goes  on  to  show  how  these 
ambiguities  have  originated.  It  is  a  matter  of  considerable 
importance  to  the  logician  to  understand  the  process,  and  we 
reproduce,  at  length,  Jevons's  treatment  of  it : 


THE  AMBIGUITY  OF  TERMS  57 

"  There  are  two  great  and  contrary  processes,"  he  says, 
"  which  modify  language,  as  follows  : 

"  1.  Generalization,  by  which  a  name  comes  to  be  apjahed  to 
a  wider  class  of  objects  than  before,  so  that  the  extension  of 
its  meaning  is  increased,  and  the  intension  diminished. 

"  2.  Specialization,  by  which  a  name  comes  to  be  restricted  to 
a  narrower  class,  the  extension  being  decreased  and  the  inten- 
sion increased.* 

"  The  first  change  arises  in  the  most  obvious  manner  from  our 
detecting  a  resemblance  between  a  new  object  which  is  without 
a  name  and  some  well-known  object.  To  express  the  resem- 
blance we  are  instinctively  led  to  apply  the  old  name  to  the 
new  object.  Thus  we  are  well  accpiainted  with  glass,  and  if  we 
meet  any  substance  having  the  same  glassy  nature  and  appear- 
ance we  shall  be  apt  at  once  to  call  it  a  kind  of  glass.  The 
word  coal  has  undergone  a  change  of  this  kind  ;  originally  it 
was  the  name  of  charked  or  charred  wood,  which  was  the  prin- 
cipal kind  of  fuel  used  five  hundred  years  ago.  As  mineral 
coal  came  into  use  it  took  the  name  from  the  former  fuel, 
which  it  resembled  more  nearly  than  anything  else,  but  was  at 
first  distinguished  as  sea-coal  or  pit-coal.  Being  now  the  far 
more  common  of  the  two,  it  has  taken  the  simple  name,  and  we 
distinguish  charred  wood  as  charcoal.  Paper  has  undergone  a 
like  change  :  originally  denoting  the  papyrus  used  in  the  Ko- 
man  empire,  it  was  transferred  to  the  new  writing  material, 
made  of  cotton  or  linen  rags,  which  was  introduced  at  a  quite 
uncertain  period.  The  word  character  is  interesting  on  ac- 
count of  its  logical  employment  ;  the  Greek  xaPaKTVP  denoted 
strictly  a  tool  for  engraving,  but  it  was  transferred  by  asso- 
ciation to  the  marks  or  letters  engraved  with  it,  and  this 
meaning  is  still  retained  by  the  word  when  we  speak  of  Greek 
characters,  Arabic  characters,  i.e.,  figures  or  letters.  But  inas- 
much as  objects  often  have  natural  marks,  signs,  or  tokens 
which  may  indicate  them  as  well  as  artificial  characters,  the 
name  was  generalized  and  now  means  any  peculiar  or  distinc- 
tive mark  or  quality  by  which  an  object  is  easily  recognized. 

*  For  explanation  of  the  terms  intension  and  extension  see  Chapter  V. 


58  ELEMENTS  OF  LOGIC 

"  Changes  of  this  kind  are  usually  effected  by  no  particular 
person  and  with  no  distinct  purpose,  but  by  a  sort  of  uncon- 
scious instinct  in  a  number  of  persons  using  the  name.  In 
the  language  of  science,  however,  changes  are  often  made  pur- 
posely, and  with  a  clear  apprehension  of  the  generalization 
implied.  Thusvsoap  in  ordinary  life  is  applied  only  to  a  com- 
pound of  soda  or  potash  with  fat,  but  chemists  have  purposely 
extended  the  name  so  as  to  include  any  compound  of  a  metal- 
lic salt  with  a  fatty  substance.  Accordingly  there  are  such 
things  as  lime-soap  and  lead-soap,  which  latter  is  employed  in 
making  common  diachylon  plaster.  Alcohol  at  first  denoted 
the  product  of  ordinary  fermentation  commonly  called  spir- 
its of  wine,  but  chemists  having  discovered  that  many  other 
substances  had  a  theoretical  composition  closely  resembling 
spirits  of  wine,  the  name  was  adopted  for  the  whole  class  and 
a  long  enumeration  of  different  kinds  of  alcohol  will  be  found 
in  Dr.  Roscoe's  lessons  on  chemistry.  The  number  of  known 
alcohols  is  likewise  subject  to  indefinite  increase  by  the  prog- 
ress of  discovery.  Every  one  of  the  chemical  terms,  acid, 
alkali,  metal,  alloy,  earth,  ether,  oil,  gas,  salt,  may  be  shown  to 
have  undergone  great  generalizations. 

"  In  other  sciences  there  is  hardly  a  less  supply  of  instances. 
A  lens  originally  meant  a  lenticular-shaped  or  double  convex 
piece  of  glass,  that  being  the  kind  of  glass  most  frequently  used 
by  opticians.  But  as  glasses  of  other  shapes  came  to  be  used 
along  with  lenses,  the  name  was  extended  to  concave  or  even 
to  perfectly  flat  pieces  of  glass.  The  words  lever,  plane,  cone, 
cylinder,  arc,  conic  section,  curve,  prism,  magnet,  pendulum, 
ray,  light,  and  many  others,  have  been  similarly  generalized. 

"  In  common  language  we  may  observe  that  even  proper  or 
singular  names  are  often  generalized,  as  when  in  the  time  of 
Cicero  a  good  actor  was  called  a  Roscius,  after  an  actor  of  pre- 
eminent talent.  The  name  Csesar  was  adopted  by  the  succes- 
sor of  Julius  Csesar  as  an  official  name  of  the  Emperor,  with 
which  it  gradually  became  synonymous,  so  that  in  the  present 
day  the  kaisers  of  Austria  and  the  czars  of  Russia  both 
take  their  title  from  Csesar.      The  celebrated  tower  built  by 


THE  AMBIGUITY  OF  TERMS  59 

the  king  of  Egypt  on  the  island  of  Pharos,  at  the  entrance  of 
the  harbor  of  Alexandria,  has  caused  light-houses  to  be  called 
jjhares  in  French,  and  pharos  in  obsolete  English.  From  the 
celebrated  Roman  general,  Quintus  Fabius  Maximus,  any  one 
who  avoids  bringing  a  contest  to  a  crisis  is  said  to  pursue  a 
Fabian  policy. 

"  In  science  also  singular  names  are  often  extended,  as  when 
fixed  stars  are  called  distant  suns,  or  the  companions  of  Jupiter 
are  called  his  moons.  It  is,  indeed,  one  theory,  and  a  probable 
one,  that  all  general  names  were  created  by  the  process  of 
generalization  going  on  in  the  early  ages  of  human  progress. 
As  the  comprehension  of  general  notions  requires  higher  intel- 
lect than  the  apprehension  of  singular  and  concrete  things, 
it  seems  natural  that  names  shoiild  at  first  denote  individual 
objects  and  should  afterward  be  extended  to  classes.  We 
have  a  glimpse  of  this  process  in  the  case  of  the  Australian 
natives,  who  had  been  accustomed  to  call  a  large  dog  caclU,  but 
when  horses  were  first  introduced  into  the  country  they  adoj^ted 
this  name  as  the  nearest  description  of  a  horse.  A  very  sim- 
ilar incident  is  related  by  Captain  Cook  of  the  natives  of 
Otaheite.  It  may  be  objected,  however,  that  a  certain  process 
of  judgment  must  have  been  exerted  before  the  suitability  of  a 
name  to  a  particular  thing  could  have  been  perceived,  and  it 
may  be  considered  probable  that  specialization  as  well  as 
generalization  must  have  acted  in  the  earliest  origin  of  lan- 
guage much  as  it  does  at  present. 

"Specialization  is  an  exactly  opposite  process  to  generaliza- 
tion, and  is  almost  equally  important.  It  consists  in  narrow- 
ing the  extension  of  meaning  of  a  general  name,  so  that  it 
comes  to  be  a  name  only  of  an  individual  or  a  minor  part  of 
the  original  class.  It  is  thus  we  are  furnished  with  the  requi- 
site names  for  a  multitude  of  new  implements,  occupations, 
and  ideas  with  which  we  deal  in  advancing  civilization.  The 
name  physician  is  derived  from  the  Greek  <£uo-ikos,  natural, 
and  <£v<x<.s,  nature,  so  that  it  properly  means  one  who  has  stud- 
ied nature,  especially  the  nature  of  the  human  body.  It  has 
become  restricted,  however,  to  those  who  use  this  knowledge 


60  ELEMENTS  OF  LOGIC 

for  medical  purposes,  and  the  investigators  of  natural  science 
have  been  obliged  to  adopt  the  term  physicist.  The  name  nat- 
uralist has  been  similarly  restricted  to  those  who  study  ani- 
mated nature.  The  name  surgeon  originally  meant  handi- 
craftsman, being  a  corruption  of  chirurgeon,  derived  from  the 
Greek  x€LP0VPv^'  handworker.  It  has  long  been  specialized, 
however,  to  those  who  perform  the  mechanical  parts  of  the 
sanatory  art. 

"Language  abounds  with  equally  good  examples.  Minister 
originally  meant  a  servant,  or  one  who  acted  as  a  minor  of 
another.  Now  it  often  means,  specially,  the  most  important 
man  in  the  kingdom.  A  chancellor  was  a  clerk,  or  even  a 
doorkeeper,  who  sat  in  a  place  separated  by  bars  or  cancelli 
in  the  offices  of  the  Roman  Emperor's  palace  ;  now  it  is  always 
the  name  of  a  high  or  even  the  highest  dignitary.  Peer  was  an 
equal  (Latin  par),  and  we  still  speak  of  being  tried  by  our 
peers  ;  but  now,  by  the  strange  accidents  of  language,  it  means 
the  few  who  are  superior  to  the  rest  of  the  Queen's  subjects  in 
rank.  Deacon,  bishop,  clerk,  queen,  captain,  general,  are  all 
words  which  have  undergone  a  like  process  of  specialization. 
In  such  words  as  telegraph,  rail,  signal,  station,  and  many 
words  relating  to  new  inventions,  we  may  trace  the  progress 
of  change  in  a  lifetime. 

"One  effect  of  this  process  of  specialization  is  very  soon  to 
create  a  difference  between  any  two  words  which  happen  from 
some  reason  to  be  synonymous.  Two  or  more  wTords  are  said 
to  be  synonymous  (from  the  Greek  avv,  with,  and  ovofia,  name) 
when  they  have  the  same  meaning,  as  in  the  case,  perhaps,  of 
teacher  and  instructor,  similarity  and  resemblance,  beginning 
and  commencement,  sameness  and  identity,  hypothesis  and 
supposition,  intension  and  comprehension.  But  the  fact  is 
that  words  commonly  called  synonymous  are  seldom  perfect- 
ly so,  and  there  are  almost  always  shades  of  difference  in 
meaning  or  use,  which  are  explained  in  such  works  as  Crabb's 
'  English  Synonyms.'  A  process  called  by  Coleridge  Desyno- 
nymization,  and  by  Herbert  Spencer,  Differentiation,  is  always 
going  on,  which  tends  to  specialize  one  of  a  pair  of  synonymous 


THE  AMBIGUITY  OF  TERMS  61 

words  to  one  meaning  and  the  other  to  another.  Thus  wave 
and  billow  originally  meant  exactly  the  same  physical  effect, 
but  poets  have  now  appropriated  the  word  '  billow,'  whereas 
wave  is  used  chiefly  in  practical  and  scientific  matters.  Un- 
dulation is  a  third  synonym,  which  will  probably  become  the 
sole  scientific  term  for  wave  in  course  of  time.  Cab  was  origi- 
nally a  mere  abbreviation  of  cabriolet,  and  therefore  of  similar 
meaning,  but  it  is  now  specialized  to  mean  almost  exclusively 
a  hackney  cab.  In  America  car  is  becoming  restricted  to 
the  meaning  of  a  railway  car. 

"It  may  be  remarked  that  it  is  a  logical  defect  in  a  language 
to  possess  a  great  number  of  synonymous  terms,  since  we  ac- 
quire the  habit  of  using  them  indifferently,  without  being  sure 
that  they  are  not  subject  to  ambiguities  and  obscure  differ- 
ences of  meaning.  The  English  language  is  especially  subject 
to  the  inconvenience  of  having  a  complete  series  of  words 
derived  from  Greek  or  Latin  roots  nearly  synonymous  with 
other  words  of  Saxon  or  French  origin.  The  same  statement 
may,  in  fact,  be  put  into  Saxon  or  classical  English  ;  and  we 
often,  as  Whately  has  well  remarked,  seem  to  prove  a  state- 
ment by  merely  reproducing  it  in  altei*ed  language.  The  rhe- 
torical power  of  the  language  may  be  increased  by  the  copious- 
ness and  variety  of  diction,  but  jntfalls  are  thus  prepared  for 
all  kinds  of  fallacies. 

"In  addition  to  the  effects  of  generalization  and  specializa- 
tion, vast  additions  and  changes  are  made  in  language  by  the 
process  of  analogous  or  metaphorical  extension  of  the  meaning 
of  words.  This  change  may  be  said,  no  doubt,  to  consist  in 
generalization,  since  there  must  always  be  a  resemblance  be- 
tween the  new  and  old  applications  of  the  term.  But  the  re- 
semblance is  often  one  of  a  most  distant  and  obscure  kind, 
such  as  we  should  call  analogy  rather  than  identity.  All  words 
used  metaphorically,  or  as  similitudes,  are  cases  of  this  process 
of  extension.  Thus  the  old  similitude  of  a  ruler  to  the  pilot 
of  the  vessel  gives  rise  to  many  metaphors,  as  in  speaking  of 
the  Prime  Minister  being  at  the  helm  of  the  state.  The  word 
governor,  and  all  its  derivatives,  is,  in  fact,  one  result  of  this 


62  ELEMENTS  OF  LOGIC 

metaphor,  being  merely  a  corrupt  form  of  gubernator,  steers- 
man. The  words  compass,  pole-star,  ensign,  anchor,  and  many 
others  connected  with  navigation,  are  constantly  used  in  a  met- 
aphorical manner.  From  the  use  of  horses  and  hunting  we  de- 
rive another  set  of  metaphors  ;  as,  in  taking  the  reins  of  gov- 
ernment, overturning  the  government,  taking  the  bit  between 
the  teeth,  the  government  whip  being  heavily  weighted,  etc. 
No  doubt  it  might  be  shown  that  every  other  important  occupa- 
tion of  life  has  furnished  its  corresponding  stock  of  metaphors. 

"It  is  easy  to  show,  however,  that  this  process,  besides  going 
on  unconsciously  at  the  present  day,  must  have  acted  through- 
out the  history  of  language,  and  that  we  owe  to  it  almost  all, 
or  probably  all,  the  words  expressive  of  refined  mental  or 
spiritual  ideas.  The  very  word  spirit,  now  the  most  refined 
and  immaterial  of  ideas,  is  but  the  Latin  spiritus,  a  gentle 
breeze  or  breathing  ;  and  inspiration,  esprit,  or  wit,  and  many 
other  words,  are  due  to  this  metaphor.  It  is  truly  curious, 
however,  that  almost  all  the  words  in  different  languages  de- 
noting mind  or  soul  imply  the  same  analogy  to  breath.  Thus, 
soul  is  from  the  Gothic  root  denoting  a  strong  wind  or  storm  ; 
the  Latin  words  animus  and  anima  are  supposed  to  be  con- 
nected with  the  Greek,  ave/xos,  wind ;  i/^x7?  *s  certainly  derived 
from  \p\>x<»,  to  blow ;  Trvev/xa,  air,  or  breath,  is  used  in  the  New 
Testament  for  Spiritual  Being  ;  and  our  word  ghost  has  been 
asserted  to  have  a  similar  origin. 

"Almost  all  the  terms  employed  in  mental  philosophy  or 
metaphysics,  to  denote  actions  or  phenomena  of  mind,  are  ul- 
timately derived  from  metaphors.  Apprehension  is  the  putting 
forward  of  the  hand  to  take  anything  ;  comprehension  is  the 
taking  of  things  together  in  a  handful ;  extension  is  the  spread- 
ing out  ;  intention,  the  bending  to  ;  explication,  the  unfolding ; 
application,  the  folding  to ;  proposition,  the  placing  before  ; 
intuition,  the  seeing  into  ;  and  they  might  be  almost  indefi- 
nitely extended.  Our  English  name  for  reason,  the  understand- 
ing, obviously  contains  some  physical  metaphor  which  has  not 
been  fully  explained  ;  with  the  Latin  intellect  there  is  also  a 
metaphor. 


THE  AMBIGUITY  OF  TERMS  63 

"  Every  sense  gives  rise  to  words  of  refined  meaning  ;  sapi- 
ence, taste,  insipidity,  goilt,  are  derived  from  the  sense  of  taste  ; 
sagacity,  from  the  dog's  extraordinary  power  of  smell  ;  but  as 
the  sense  of  sight  is  by  far  the  most  acute  and  intellectual,-  it 
gives  rise  to  the  larger  part  of  language ;  clearness,  lucidity, 
obscurity,  haziness,  perspicuity,  and  innumerable  other  expres- 
sions, are  derived  from  this  sense. 

"It  is  truly  astonishing  to  notice  the  power  which  language 
possesses  by  the  processes  of  generalization,  specialization,  and 
metaphor,  to  create  many  words  from  one  single  root.  Pro- 
fessor Max  Midler  has  given  a  remarkable  instance  of  this  in 
the  case  of  the  root  spec,  which  means  sight,  and  appears  in  the 
Aryan  languages,  as  in  the  Sanscrit  spas,  the  Greek  a kIttto /*cu, 
with  transposition  of  consonants  in  the  Latin  specio,  and  even 
in  the  English  spy.  The  following  is  an  incomplete  list  of  the 
words  developed  from  this  root :  Species,  special,  especial, 
specimen,  spice,  spicy,  specious,  specialty,  specific,  specializa- 
tion, specie  (gold  or  silver),  spectre,  specification,  spectacle, 
spectator,  spectral,  spectrum,  speculum,  specular,  speculations. 
The  same  root  also  enters  into  composition  with  various  pre- 
fixes ;  and  we  thus  obtain  a  series  of  words,  suspect,  aspect, 
circumspect,  expect,  inspect,  prospect,  respect,  retrospect,  in- 
trospection, conspicuous,  perspicuous,  perspective  ;  with  each 
of  which,  again,  a  number  of  derivatives  is  connected.  Thus 
from  suspect  we  derive  suspicion,  suspicable,  suspicious,  sus- 
piciousness. I  have  estimated  that  there  are  in  all  at  least 
two  hundred  and  forty-six  words  employed  at  some  period  or 
another  in  the  English  language  which  undoubtedly  come  from 
the  root  spec."  * 

Jevons's  discussion  suffices  to  illustrate  quite  fully  the  fact 
of  ambiguous  conceptions  and  the  laws  of  their  development, 
but  it  does  not  indicate  those  instances  which  are  likely  to 

*  For  a  more  complete  study  of  the  ambiguity  of  terms  and  the  origin 
of  it.  the  student  may  consult  the  following  works:  Locke:  Essay  on 
Human  Understanding,  Book  III.,  Chapters  IX.  and  X.  ;  Mill:  Logic, 
Book  IV.,  Chapter  V.  ;  Trench  :  On  the  Study  of  Words  ■  Max  Midler  : 
Lectures  on  the  Science  of  Language. 


64  ELEMENTS  OF  LOGIC 

give  trouble  in  the  study  of  Logic.  It  is  perhaps  just  as  well 
that  we  should  first  understand  the  wide  extent  to  which  words 
vary  in  meaning,  and  to  which  they  are  modifiable  by  general- 
ization and  specialization,  before  considering  more  particular 
classes  affecting  logical  problems.  We  can  in  that  way  have 
the  general  laws  regulating  or  causing  their  ambiguity  most 
distinctly  impressed  upon  our  mind,  inasmuch  as  they  affect 
the  meaning  of  terms  even  where  logical  confusion  may  not  be 
the  consequence  of  the  change.  Most  of  the  conceptions 
chosen  by  Jevons  in  illustration  would  very  seldom  give  rise 
to  any  serious  fallacy  in  reasoning,  as,  in  spite  of  a  certain 
kind  of  ambiguity,  they  are  well  enough  understood  to  prevent 
serious  logical  mishaps.  But  this  is  not  the  case  with  a  very 
large  set  of  conceptions  current  in  science  and  philosophy, 
general  conceptions  whose  technical  meaning  varies  with  the 
schools  arguing  for  or  against  certain  doctrines  involved  in 
them.  They  belong  largely  to  the  class  of  terms  which  are 
either  very  abstract  in  their  import,  or  are  liable  to  the  con- 
fusion of  their  abstract  with  their  concrete  conception.  This 
is  a  source  of  ambiguity  already  touched  upon,  and  which 
Jevons  does  not  treat  of.  It  is  perhaps  a  source  of  more  logi- 
cal errors  than  all  the  ambiguities  his  discussion  illustrates, 
although  it  comes  under  the  same  laws  as  those  which  he  does 
notice.  Pure  abstracts  and  pure  concretes,  as  adjectival  nouns 
and  proper  or  singular  names,  are  not  likely  to  give  much 
difficulty.  Fallacies  are  much  more  incident  to  mixed  abstract 
and  concrete  conceptions,  or  those  concrete  terms,  usually  so- 
called,  whose  different  meanings  are  either  closely  allied,  or  con- 
nected with  closely  resembling  objects  having  nevertheless  im- 
portant differences  between  them.  Where  the  separate  concrete 
meanings  of  the  same  term  have  no  natural  affiliation,  they  will 
not  easily  give  rise  to  error.  Thus  "  spirit,"  denoting  mind  on 
the  one  hand,  and  alcohol  on  the  other,  will  not  be  used  in 
connections  where  fallacy  will  be  the  consequence  of  such  a 
possible  ambiguity.  But  "spirit,"  denoting  mind  or  intelli- 
gence in  one  case,  and  something  immaterial  in  another,  may 
be  the  source  of  error,  because  the  two  conceptions  are  closely 


THE  AMBIGUITY  OF  TERMS  65 

related.  The  process  of  generalization  gives  rise  to  a  tendency 
away  from  the  concrete  and  may  produce  confusion  as  long  as 
the  new  and  old  conceptions  are  not  clearly  defined.  An  op- 
posite source  of  error  is  that  of  the  process  of  specialization, 
which  is  from  the  more  abstract  to  the  concrete  ;  in  so  far  as 
the  term  "  abstract "  is  taken  to  denote  the  common  cpialities 
denoted  by  the  general  concept  apart  from  the  particular,  con- 
crete, and  differentiated  individual  to  which  it  may  also  apply. 
The  generalization  and  specialization  indicated  have  to  do, 
almost  exclusively,  with  the  terms  that  may  be  mixed  abstracts 
and  concretes,  and  the  source  of  confusion  increases  as  the 
indefiniteness  of  a  conception  increases  with  the  process  of 
generalization. 

A  highly  illustrative  set  of  ambiguous  terms,  more  impor- 
tant to  scientific  and  philosophic  study  than  any  which  Jevons 
has  stated,  and  which  yet  come  under  the  laws  which  he 
enunciates,  are  such  as  the  following  :  Motive,  intuition,  expe- 
rience, idea,  cause,  God,  religion,  faith,  feeling,  knowledge,  sen- 
sation, reason,  first  principles,  a  priori,  government,  law,  nature, 
moral,  right,  justice,  nation,  church,  authority,  origin,  freedom, 
etc.  Thus  motive  may  denote  in  mechanics  the  cause  or  force 
producing  motion,  and  by  analogy  the  cause  of  volition  ;  then 
it  may  denote  the  idea  of  an  end,  which,  in  so  far  as  it  is  the 
object  of  the  mind,  is  rather  the  effect  than  the  cause  of  voli- 
tion. Intuition  may  denote  immediate  perception,  or  again 
it  may  denote  universal  perception,  the  first  indicating  only 
a  direct  process  of  knowledge  without  reference  to  time  or 
the  number  of  persons  involved,  and  the  second  indicating 
that  all  individuals  experience  it.  The  term  has,  further, 
four  or  five  other  meanings,  but  these  two  suffice  to  illustrate 
an  ambiguity  which  might  well  give  rise  to  all  the  contro- 
versies waged  about  certain  doctrines  in  philosophy.  Add 
to  this  the  still  greater  ambiguity  of  the  terms  "experience" 
and  "idea,"  and  we  can  well  imagine  why  so  much  dispute 
has  centred  about  the  theory  of  "  innate  ideas."  On  the 
one  hand,  experience  has  meant,  first,  simple  sensations  ;  sec- 
ond, any  realized  state  of  consciousness  ;  third,  a  series  of  men- 


66  ELEMENTS  OF  LOGIC 

tal  states  giving  as  a  resultant  a  new  component  not  found 
in  the  primary  state  ;  on  the  other  hand,  idea  has  meant,  first, 
simple  presentations  of  sense  (Locke)  ;  second,  general  concep- 
tions which  are  the  product  of  the  higher  intellectual  faculties ; 
and  third,  mere  opinion.  These  are  sufficient  to  make  a  per- 
fect labyrinth  of  complexities  in  argument.  Again,  cause  is 
sometimes  merely  the  condition  of  anything,  and  at  others  the 
active  agent  producing  a  phenomenon.  In  this  latter  sense  it 
may  be  either  some  influence  external  to  the  thing  affected,  or, 
as  in  the  case  of  the  mind  and  its  volition,  or  of  any  agent 
which  contributes  of  its  own  energy  to  the  effect,  it  may  be 
internal.  God  sometimes  denotes  the  first  cause  without  in- 
dicating whether  it  is  more  than  force  ;  then  it  may  denote 
the  agent  effecting  the  order  of  the  world,  whether  regarded 
or  not  as  the  creator  of  matter  ;  again  the  term  may  denote  a 
supreme  intelligence,  with  various  attributes  of  perfection  and 
power,  not  necessarily  implied  in  the  former  conceptions,  but 
not  excluded  by  them.  Religion  may  denote  certain  beliefs 
about  God  and  the  world,  or  it  may  denote  a  certain  attitude 
of  mind  toward  these  things,  or  it  may  further  denote  the  be- 
lief and  practice  of  certain  moral  doctrines,  with  various  sub- 
ordinate meanings.  Faith  has  at  least  four  distinct  meanings  : 
First,  intellectual  assent  to  propositions  above  the  attestation 
of  reason,  and  thus  equivalent  to  intuition,  if  it  gives  immedi- 
ate assured  knowledge,  or  to  mere  probability,  if  it  can  only 
produce  less  than  absolute  conviction  or  certitude  ;  second,  it 
may  denote  the  acceptance  of  truth  on  authority  ;  third,  fidel- 
ity of  disposition  in  living  up  to  a  promise,  treaty,  or  a  law  ; 
and  fourth,  trust  in  a  person.  Feeling  has  a  similar  applica- 
tion, now  denoting  a  tactual  sensation,  now  a  firm  and  ineradi- 
cable conviction  =  intuition  ;  again,  a  variable  mental  state  of 
excitement  which  can  easily  be  eradicated  from  an  influence 
upon  knowledge,  and  so  is  equivalent  to  emotion,  and  lastly,  a 
general  conception  for  the  primary  elements  of  knowledge. 
The  remaining  terms  and  many  others  possess  similar  multi- 
plications of  meaning,  which  are  the  source  of  all  the  contro- 
versies and  their  incident  fallacies  in  the  world  of  speculative 


THE  AMBIGUITY  OF  TEEMS  67 

knowledge.  The  few  that  we  have  specified  may  serve  to  illus- 
trate the  importance  of  being  on  the  alert  for  them,  and  of 
first  giving  a  term  that  analysis  or  definition  of  its  meaning 
which  is  at  least  a  partial  provision  against  the  contingency  of 
error. 

To  recapitulate.  First,  words  are  ambiguous  when  they 
are  capable  of  more  than  a  single  meaning.  Second,  they  be- 
come ambiguous  through  the  process  of  transition  from  a 
generalized  to  a  specialized  form,  or  the  reverse.  Third,  this 
process  causes  confusion  in  Logic  mainly  when  it  results  in 
the  liability  to  mistake  the  abstract  for  the  concrete,  or  one 
concrete  conception  for  another  closely  allied  to  it. 


CHAPTER  V. 

THE   INTENSION   AND   THE   EXTENSION   OF   CONCEPTS 

1st.  Nature  of  Intension  and  Extension. — Allusion  has 
been  made  to  the  intension  and  the  extension  of  conceptions 
without  explaining  the  meaning  of  those  terms.  We  come 
now  to  determine  this  meaning,  which  is  a  very  important 
matter  in  Logic.  Various  terms  have  been  employed  to  ex- 
press the  same  fact  and  relation  as  are  expressed  by  extension 
and  intension.  Thus  comprehension,  depth,  connotation  are 
frequently  taken  as  synonymous  with  intension,  and  extent, 
breadth,  denotation  for  the  extension  of  terms.  Much  contro- 
versy exists  about  the  true  use  of  the  terms  denotation  and 
connotation,  which  I  shall  consider  later  in  the  chapter.  The 
only  matter  of  real  importance  is  the  meaning  of  the  concepts 
which  they  are  supposed  to  describe,  and  this  can  be  deter- 
mined, in  a  large  measure  at  least,  without  complicating  our- 
selves with  this  controversy. 

Nearly  all,  if  not  absolutely  all  terms  have  a  twofold  mean- 
ing or  application,  which  is  expressed  by  their  intension  and 
extension.  A  simple  example  will  make  this  apparent.  The 
term  "  man  "  may  apply  to  all  the  individual  men  represented 
by  the  word.  It  is  thus  a  name  for  the  individuals  in  the 
class,  and  we  should  call  each  one  by  that  name  when  asked  to 
define  what  he  is.  On  the  other  hand,  the  term  also  expresses 
a  certain  number  of  qualities  or  marks  which  make  up  the  in- 
dividual. "  Man  "  is  thus  not  only  a  name  for  the  individuals 
of  the  class,  but  for  a  certain  conjunction  of  qualities,  which 
may  be  thought  of  without  regard  to  the  range  of  application 
possessed  by  the  term.  Tree  has  a  similar  application.  It 
may  be  a  name  for  a  class  of  objects,  or  it  may  denote  a  cer- 
tain number  of  vegetable  qualities.     Metal,  quadruped,  biped, 


INTENSION  AND  EXTENSION  OF  CONCEPTS        69 

vertebrate,  triangle,  figure,  nation,  city,  custom,  etc.,  are  only 
other  instances  of  the  same  fact.  There  are  some  terms,  how- 
ever, which  are  not  class  concepts  ;  for  example,  singular  or 
proper  names.  They  seem  only  to  denote  a  combination  of 
qualities,  and  it  is  true  that  they  cannot  denote  more  than  one 
individual.  But  this  does  not  hinder  it  from  having  a  numer- 
ical application  to  that  extent,  and  this  is  all  that  is  necessary 
to  justify  the  application  of  extension  to  it.  In  all  these  terms 
the  intension  refers  to  the  quality  or  qualities  possessed  by  an 
object  having  a  given  name  ;  the  extension  refers  to  the  number 
of  objects  included  under  the  name.  These  will  at  least  serve 
as  approximate  definitions  of  the  two  terms  until  completed. 
We  require  at  present  merely  to  know  that  the  extension  indi- 
cates the  objects  to  which  a  name  applies,  and  the  intension, 
the  attributes  which  it  imjilies. 

The  clearest  illustration  of  terms  with  extension  will  be 
class  concepts,  and  especially  all  concrete  general  concepts,  as 
"  man,"  "  quadruped,"  "  Caucasian,"  "  European,"  "  tree," 
"book,"  "animal,"  "house,"  etc.,  and  perhaps  the  clearest  il- 
lustration of  intension  will  be  pure  abstract  terms  and  singular 
or  proper  names,  which  latter,  although  they  denote  an  indi- 
vidual in  a  class,  more  particularly  indicate  a  certain  quality 
or  union  of  qualities  that  are  thought  of  rather  than  the  range 
of  the  term,  as  "  Lincoln,"  "  Berlin,"  "  Rome,"  "  Europe," 
"Plato,"  "Declaration  of  Independence,"  "Magna  Charta," 
etc.  But  all  terms  have  both  references  at  the  same  time,  and 
differ  only  in  the  degree  of  their  intension  and  extension  in 
relation  to  each  other.  That  is,  every  term  has  both  intension 
and  extension.  The  extension  of  "man  "  is  the  number  of  in- 
dividuals to  which  it  is  applicable  as  a  name  ;  the  intension  is 
the  number  of  qualities  which  it  implies,  and  so  with  other 
terms. 

The  application  of  extension  and  intension  to  concrete  con- 
ceptions, singular  or  general,  affords  no  difficulty.  But  is  it 
possible  to  apply  them  to  abstract  terms?  The  answer  to 
this  question  involves  the  previous  questions  whether  abstract 
terms  are  singular  or  general,  and  whether  singular  terms  can 


70  ELEMENTS  OF  LOGIC 

be  said  to  have  extension.  Inasmuch  as  I  have  claimed  sin- 
gular concepts  for  extension,  merely  indicating  that  the  ex- 
tension is  at  its  minimum  in  them,  there  remains  only  to  settle 
whether  abstract  terms  are  singular  or  general.  Jevons  thinks 
them  singular.  This  may  possibly  be  the  case  in  some  in- 
stances. But  it  is  certainly  not  the  case  in  such  terms  as 
"  sweetness,"  "justice,"  "  ability,"  "  color,"  etc.  For  there  are 
several  kinds  of  "  sweetness,"  several  divisions  of  "justice," 
and  various  forms  of  "ability"  and  "color."  Perhaps  some 
writers  would  make  "color  "  a  concrete  concept.  I  hold  that 
it  may  be  either  concrete  or  abstract,  according  to  its  refer- 
ence. But  it  is  not  important  to  settle  this  question,  whether 
it  may  be  both,  or  is  only  one  of  them.  If  it  be  concrete  it 
certainly  has  extension  and  intension  together,  and  if  it  be 
abstract  and  general,  as  it  must  be,  both  extension  and  in- 
tension are  characteristic  of  it ;  so  that  the  two  qualities  can 
be  denied  of  it  only  on  the  supposition  that  it  is  abstract  and 
singular,  and  that  all  singulars  have  only  intension.  But  with 
the  qualification  already  mentioned  these  properties  are  pos- 
sessed by  both  concrete  and  abstract  terms,  so  that  the  re- 
duction of  abstract  concepts  to  singulars  would  be  no  obstacle 
to  their  simultaneous  possession  of  intension  and  extension. 

It  remains  to  consider  whether  adjectives  can  possess  in- 
tension and  extension.  In  so  far  as  they  are  the  names  of 
qualities  they  indicate  some  degree  of  intension.  But  are 
they  class  terms?  There  are  two  ways  of  answering  this 
question.  The  first  is  Mill's  view  that  attributive  terms  of 
this  kind  always  imply  a  subject.  Thus  "white"  implies  all 
white  objects,  and  so  must  represent  a  class.  The  second 
is  the  view  that,  even  if  they  do  not  necessarily  imply  anything 
definite  about  things,  they  may  connote  different  kinds  or 
degrees  of  the  quality  expressed  by  them.  Thus  there  are 
many  shades  of  "white,"  "blue,"  "green;"  many  kinds  or  de- 
grees of  "pure,"  "noble,"  " benevolent,"  etc.  In  this  sense 
they  will  also  have  extension.  And  also  they  would  possess 
this  quality  if  they  were  singular,  according  to  the  principle  I 
have  asserted  for  the  minimum  extension  of  a  term,  and  it  is 


INTENSION  AND  EXTENSION  OF  CONCEPTS         71 

not  necessary,  perhaps,  to  take  account  of  any  other  degree 
of  it. 

It  may  also  be  a  question  whether  any  great  importance 
attaches  to  either  the  affirmation  or  the  denial  of  extension  to 
abstract  terms  and  attributives.  The  chief  interest  of  the 
logician,  probably,  is  in  the  possibility  of  representing  them 
by  the  usual  symbols,  the  logical  circles,  in  illustrating  the 
relation  between  the  subject  and  predicate  of  a  proposition- 
But  as  this  symbolization  is  only  one  of  analogy,  and  as 
the  legitimacy  of  it  even  is  disputed  by  some,  we  may  disre- 
gard the  question  whether  it  be  strictly  applicable  to  abstract 
nouns  and  to  adjectives,  and,  if  we  desire,  leave  the  whole 
matter  undecided,  so  far  as  practical  Logic  is  concerned,  be- 
cause fallacies  do  not,  to  any  extent,  turn  upon  the  question 
whether  abstract  terms  and  adjectives  are  capable  of  extension 
or  not.  In  purely  theoretical  Logic  it  may  be  somewhat 
different.  There,  we  may  be  interested  merely  in  the  truth 
about  this  special  case,  and  undoubtedly  we  shall  find  the 
matter  of  extension  less  clear  in  its  application  to  abstract 
terms  and  adjectives,  even  if  it  be  granted  as  possible,  than 
to  concrete  substantives.  For,  being  terms  which  merely 
imply  a  subject,  but  do  not  express  it  definitely,  they  will  most 
likely  represent  either  a  single  quality  with  its  various  degrees, 
or  only  such  as  the  substantive  from  which  they  are  taken 
represents.  Of  course  some  adjectives  are  not  taken  from 
substantives.  But  many  of  those  that  are  so  taken,  as 
"manly,"  "human,"  "animal,"  "personal,"  "heavenly,"  un- 
doubtedly connote  as  many  qualities  as  the  substantive,  and  so 
equal  it  in  its  intension  at  least,  and  it  is  possible  to  say  that 
their  extension  is  also  equal  to  that  of  the  terms  from  which 
they  are  derived ;  although  it  might  be  proper  to  regard  it  as  a 
sort  of  relative  or  derivative  extension.  Certainly  they  do  not  as 
terms  indicate  individuals  so  distinctly  as  substantive  concretes. 
But  even  if  they  do  not,  Logic  does  not  require  us  to  give  them 
more  than  the  minimum  extension,  and  this  will  apply  to  all 
concepts  representing  any  idea  whatever.  This  is  to  say  that 
extension  must  not  be  confused  with  the  notion  of  plurality. 


72  ELEMENTS  OF  LOGIC 

After  this  discussion  we  may  conclude  the  section  with  a 
more  accurate  definition  of  extension  and  intension.  The  for- 
mer is  usually  described  as  referring  to  the  individuals  of  a 
class,  and  this  conception  of  it  leaves  the  impression  that  in 
order  to  have  extension  at  all  a  term  must  be  a  class  concept. 
But  this  is  only  an  incident  of  the  disproportion  that  may  ex- 
ist between  the  two  properties.  But  to  cover  the  case  more 
distinctly  they  may  be  defined  as  follows  : 

Extension  is  the  quantitative  power  of  terms  or  concepts,  and 
so  indicates  their  numerical  application.  It  may  refer  to  in- 
dividual- or  class- wholes,  whether  substantive  or  attributive, 
real  or  conceptual. 

Intension  is  the  qualitative  power  of  terms  or  concepts,  and  so 
indicates  their  denotation  of  qualities.  It  may  refer  to  a 
single  quality,  a  group  of  qualities  in  an  individual- whole,  or 
the  common  quality  or  qualities  of  a  class-whole,  whether  sub- 
stantive or  attributive,  real  or  conceptual. 

Conceptual-wholes  are  such  as  denote  thought  products 
which  are  not  conceived  properly  as  either  substantive  or  at- 
tributive. Illustrations  are  such  concepts  as  "proposition," 
"  word,"  "  syllogism,"  "  science,"  "  botany,"  etc.  Some  terms 
describing  mental  states  may  be  regarded  as  attributive  ;  e.g., 
"  sensation,"  "  memory,"  etc. 

2d.  The  Relation  between  Extension  and  Intension. 
— The  relation  between  the  intension  and  the  extension  of  con- 
cepts is  determined  by  comparing  the  broader  with  the  narrower 
term.  Thus,  if  we  take  the  term  "  metal "  and  compare  it  with 
the  term  "  iron  "  we  shall  find  that  "  metal "  is  a  name  for  a 
larger  number  of  objects  than  the  term  "  iron,"  because  it  in- 
cludes all  that  is  denoted  by  "iron,"  and  all  other  metals  be- 
sides. The  extension  of  metal  is,  therefore,  said  to  be  greater 
than  the  extension  of  "  iron."  But  the  intension  of  "  iron  "  is 
greater  than  that  of  "  metal,"  because  it  contains  all  the  quali- 
ties necessary  to  regard  it  as  a  "metal,"  and  in  addition  the 
quality  or  qualities  necessary  to  make  it  "  iron  "  and  to  dis- 
tinguish it  from  other  metals,  such  as  gold,  silver,  lead,  etc. 
Again,  "matter"  will  have  greater  extension  than  "metal,"  and 


INTENSION  AND   EXTENSION  OP  CONCEPTS        73 

"  steel  "  less  than  "  iron."  But  the  intension  of  "  steel "  will 
be  greater  than  that  of  "iron,"  and  the  intension  of  "matter" 
less  than  that  of  "  metal."  The  same  comparison  can  be  in- 
stituted between  any  set  of  related  terms,  such  as  biped, 
man,  European,  Frenchman,  Louis  XTV.,  or  vertebrate,  quad- 
ruped, horse,  Bucephalus  ;  or  figure,  quadrilateral,  square. 

From  this  we  deduce  the  general  law  that  as  the  extension  in- 
creases the  intension  decreases,  and  vice  verm.  The  same  law  is 
sometimes  expressed  in  a  different  way.  Thus,  extension  and 
intension  can/  in  an  inverse  ratio  to  each  other,  or  they  are  in- 
versely related  to  each  other.  Hamilton  represented  the  rela- 
tion by  a  cone  or  pyramid,  in  which  the  apex  indicated  the 
least  intension  or  extension,  as  the  case  might  be,  and  the  base 
the  greatest.  We  may  thus  symbolize  the  relation  for  a  series 
of  terms,  and  we  may  indicate  botli  the  order  of  greatest  and 
least  extension,  the  greatest  and  least  intension,  and  their  re- 
ciprocal relation  inversely  considered.  The  following  figures 
will  represent  them  : 

Extension.  Intension.  Intension  and  Extension. 


Fig.  1  Fig.  2.  Fig.  3. 

Fig.  1  represents  the  order  of  least  and  greatest  extension, 
beginning  with  the  apex  of  the  triangle  or  cone.  "Plato  "  has 
the  least  extension  of  the  series  and  applies  to  only  one  indi- 
vidual. "Vertebrate  "  has  the  greatest  extension  of  the  series, 
including  all  the  others  and  all  other  beings  having  a  certain 
anatomical  structure.  In  Fig.  2  the  order  is  reversed.  "  Ver- 
tebrate "  has  the  least  intension  because  it  stands  for  the 
fewest  qualities,  and  "  Plato  "  the  greatest  because  it  denotes 
the  largest  number  of  qualities.  Fig.  3  represents  the  recip- 
rocal relation  of  the  two  properties.  "  Plato  "  has  at  the 
same  time  the  greatest  intension  and  the  least  extension,  and 


74  ELEMENTS  OF  LOGIC 

"vertebrate"  the  greatest  extension  and  the  least  intension. 
The  intermediate  terms  vary  between  these  extremes  by  an  in- 
determinate ratio,  but  are  presumably  in  the  same  relation. 

We  must  bear  in  mind,  however,  that  the  formula  for  this 
relation  is  not  strictly  accurate.  It  is  not  true  that  the  ex- 
tension always  increases  as  the  intension  increases.  Some 
logicians,  in  their  objections  to  the  law  thus  enunciated,  go  so 
far  as  to  say  that  the  relation  in  some  instances  may  be  re- 
versed, so  that  the  intension  would  increase  as  the  extension 
decreases.  This,  however,  is  an  exceptional  state  of  the  case 
and  may  be  dismissed  from  consideration  of  the  general  rule. 
The  law  is,  perhaps,  not  meant  by  any  logician  to  be  absolute- 
ly and  universally  true  in  the  strict  sense  in  which  it  is  some- 
times expressed.  The  formula  is  a  convenient  one  for  indi- 
cating a  relation  sometimes  strictly  true,  and  sometimes  true 
with  cpialifications.  It  is  enunciated  in  mathematical  terms 
for  the  sake  of  clearness  rather  than  because  it  is  literally  true 
in  all  instances  of  comparison,  although  in  the  ideal  logical 
world  it  might  be  so.  Nevertheless,  the  absolute  and  univer- 
sal application  of  the  law,  as  we  have  formulated  it,  is  subject 
to  the  following  limitations  : 

1.  The  law  cannot  be  interpreted  in  any  strict  mathematical 
sense.  It  is,  for  instance,  not  true  that  when  the  intension  is 
doubled  the  extension  is  halved.  The  number  of  individuals 
may  even  be  increased  without  decreasing  the  intension  of 
either  the  class  term  or  of  the  individual  under  it.  Thus  I 
may  increase  the  number  of  persons  to  whom  the  term  "  man  " 
is  applicable,  and  still  not  alter  the  quantity  of  intension  repre- 
sented  by  the  term. 

Some  logicians  might  reply  that,  in  fact,  this  never  takes 
place  ;  that  every  individual  added  to  a  class  differs  by  some 
mark  from  all  others,  and  so  may  decrease  the  intension  of  the 
general  term  in  the  same  proportion,  at  least,  that  its  exten- 
sion is  increased.  This  is  conceivably  the  case,  and,  if  a  fact, 
would  sustain  the  law  in  its  general  sense,  although  it  might 
not  prove  its  strict  or  definite  mathematical  interpretation  in 
terms  of  any  specified  ratio.     Besides,  the  law  is  perhaps  de- 


INTENSION  AND  EXTENSION  OF  CONCEPTS        75 

fensible  as  a  general  formula  in  the  same  sense  in  which  psy- 
chologists speak  of  the  inverse  ratio  between  sensation  and 
perception,  or  between  the  consciousness  of  feeling  and  the 
consciousness  of  an  object,  where  we  wish  merely  to  express 
the  fact  that  the  two  do  not  vary  together  in  the  same  way, 
but  that  as  one  becomes  more  distinct  the  other  becomes  less 
so.  This  mode  of  expression  may  be  applicable  to  the  varia- 
tions, at  least  as  a  general  rule,  between  extension  and  inten- 
sion, and  be  relatively  true  when  one  increases  without  the 
other  decreasing,  and  absolutely  ti-ue,  although  not  by  any 
assignable  ratio,  when  the  increase  of  the  one  is  accompanied 
by  a  corresponding  decrease  of  the  other.  Nevertheless,  the 
variation  is  so  irregular  that  the  law  has  only  a  conventional 
value  in  the  form  in  which  it  is  usually  enunciated. 

Perhaps  a  distinction  between  two  kinds  of  general  terms 
would  enable  us  to  formulate  the  law  in  different  ways,  one  of 
them  to  suit  its  simple  mathematical  conception,  and  the  other 
to  suit  a  less  definite  conception  of  it.  It  is  a  fact  that  general 
terms  or  concepts  are  of  two  distinct  kinds,  although  the  dis- 
tinction is  not  explicitly  recognized  by  logicians.  The  two 
divisions  I  shall  call  mathematical  generals  and  logical  generals. 
The  latter  term  is  perhaps  an  unfortunate  one,  because  all 
generals  are  logical  in  the  broadest  sense.  But  I  defend  its 
use  in  the  technical  sense  to  be  defined  as  the  only  resort 
at  my  command  for  expressing  the  notion  I  have  of  the 
terms  described  by  it.  By  mathematical  general  concepts  I 
mean  those  which  are  absolutely  alike  in  their  content  or  in- 
tension, or  such  as  are  grouped  together  under  the  same  name 
solely  on  account  of  their  numerical  value.  In  such  cases  the 
extension  may  be  increased  either  without  altering  the  in- 
tension, or  in  connection  even  with  an  increase  of  the  inten- 
sion, supposing,  of  course,  that  the  increase  of  the  intension 
was  the  same  in  all  individuals  of  the  class.  Thus  if  I  assume 
a  number  of  gold  coins  exactly  alike,  I  may  add  to  the  number 
denoted  by  a  particular  name  any  number  of  like  coins  with 
exactly  the  same  qualities,  and  here  I  increase  the  extension 
without  increasing  the  intension.     And  again,  if  in  adding  a 


76  ELEMENTS  OF  LOGIC 

new  individual  to  the  class,  I  discover  a  quality  not  known  be- 
fore, but  which  is  found  to  belong  to  all  members  of  the  class, 
I  have  increased  the  intension  of  the  concept  (not  of  the  thing) 
in  the  same  proportion  as  I  have  increased  the  extension.  If 
the  new  individual  has  a  new  property  not  in  the  others  of  the 
same  class,  the  intension  of  the  general  term  or  genus  may 
remain  the  same,  or  fixed,  while  the  extension  is  increased. 
Hence  wherever  the  addition  to  the  general  concept  is  a  purely 
numerical  or  mathematical  one  the  law  of  inverse  ratio  be- 
tween extension  and  intension  does  not  hold  in  its  strict  sense, 
and  can  only  be  taken  as  a  general  statement  of  the  indepen- 
dence of  each  other  in  their  variations. 

But  in  the  case  of  the  second  class,  namely,  logical  concepts, 
the  ratio  of  variation  between  extension  and  intension  may 
more  definitely  accord  with  the  statement  of  the  law.  By  a 
logical  general  concept  I  mean  one  which  does  not  apply  in 
exactly  the  same  sense  to  the  individuals  or  species  which  it 
comprehends,  or  which  strictly  connotes  only  the  common 
qualities  of  a  class,  and  does  not  denote  the  differences.  The 
difficulty,  however,  in  strictly  defining  them  is  that  they  are 
usually  applied  in  a  mathematical  sense  at  the  same  time,  and 
hence  we  may  have  no  concepts  which  denote  only  the  com- 
mon qualities  of  a  class  without  enumerating  the  individuals 
mathematically  at  the  same  time.  This  is  the  case  with  the 
mixed  concrete  and  abstract  terms.  But  such  a  general  con- 
cept, if  it  actually  exist  in  its  purity,  will  be  such  as  compre- 
hends individuals  and  species  not  taken  merely  in  an  additive 
sense,  but  as  denoting  certain  common  qualities  allowing  great 
variations  and  differences  in  all  other  characteristics.  Thus 
"  vertebrate  "  is  a  general  concept  comprehending  individuals 
and  species,  with  such  differences  as  do  not  distinguish  one  of 
them  from  another  as  comprehended  in  the  general  concept. 
It  is  the  same  with  the  general  term  "  animal."  It  denotes  a  cer- 
tain common  quality  or  qualities  which  permit  all  sorts  of  other 
differences  without  disturbing  or  confusing  the  application  of 
the  term.  In  all  such  cases  the  increase  of  the  extension  of  the 
concept  may  be  either  mathematical,  or  both  mathematical  and 


INTENSION  AND  EXTENSION  OF  CONCEPTS        77 

logical.  In  case  it  is  mathematical,  or  the  mere  addition  of  an 
individual  exactly  like  those  already  denoted  by  the  term,  the 
law  is  subject  to  the  limitations  mentioned  in  the  instances  of 
mathematical  generals.  But  if  the  increase  is  due  to  the  ad- 
dition of  a  new  species,  presenting  new  characteristics  com- 
pared with  those  denoted  before,  the  term  is  generalized  logi- 
cally as  well  as  mathematically,  and  its  intension  is  decreased 
with  its  increase  of  extension.  Thus  the  term  "  crow  "  for  a 
long  time  denoted  a  certain  class  of  birds  with  a  black,  glossy 
plumage.  The  color  of  their  feathers  was  a  common  quality, 
being  thus  involved  in  the  intension  of  the  class.  But  as  soon 
as  "  crows  "  with  a  white  or  gray  color  were  discovered,  this 
intension  was  decreased  by  throwing  the  quality  of  color  out 
of  regard  as  a  distinctive  characteristic,  and  the  extension  was 
increased  in  proportion  to  the  new  additions  to  the  class.  The 
generalization  is  accomplished  by  taking  into  account  fewer 
common  qualities,  and  hence  the  ratio  between  the  intension 
and  the  extension  varies  inversely  in  all  such  cases.  Again, 
the  term  "  Frenchman  "  ordinarily  denotes  a  person  with  cer- 
tain race  characteristics,  and  born  in  France.  But  we  often 
find  it  used  to  denote  persons  having  only  the  race  qualities 
and  born  in  any  other  locality.  "  Hebrew  "  once  denoted  the 
inhabitants  of  Palestine.  It  now  denotes  a  race  without  any 
implications  as  to  country.  "Book"  was  once  synonymous 
with  "  beechen  boards,"  having  printed  or  written  matter  upon 
them.  It  is  now  broadened  to  denote  any  mass  of  paper  bound 
in  a  particular  way,  whether  containing  printed  and  written 
matter  or  not.  The  matter,  indeed,  is  an  entirely  unessential 
characteristic,  as  it  was  once  the  essential.  Here  again  the  law 
is  strictly  illustrated,  as  the  intension  decreases  with  an  increase 
of  the  extension.  The  same  fact  is  clearly  illustrated  in  all 
cases  of  generalization  and  specialization  already  discussed. 
Here  the  intension  and  extension  vary  inversely. 

But  it  is  important  to  remark  the  limitation  with  which  this 
is  true.  The  addition  to  the  general  term  numerically  may  be 
manifold  as  great  as  the  deduction  from  its  intension.  A  hun- 
dred, more  or  less,  individuals  mav  be  added  to  the  extension 


78  ELEMENTS  OF  LOGIC 

and  only  one  or  two  qualities  subtracted  from  the  intension. 
And  even  if  only  one  individual  with  a  difference  is  added  to 
the  class,  and  two  or  more  qualities  subtracted  from  the  inten- 
sion, the  law  is  still  true  in  the  sense  it  was  intended,  although 
not  true  in  the  mathematical  sense  that  the  ratio  of  variation  is 
the  same  for  both  intension  and  extension.  In  its  proper  logi- 
cal meaning  the  law  simply  indicates,  without  any  numerical 
implications,  that  as  you  extend  the  application  of  a  term  to 
new  species  you  decrease  the  number  of  generic  qualities  com- 
prehended by  it.  This  will  be  quite  uniformly  true  of  logical 
extension,  but  variable  and  subject  to  modifications  in  mathe- 
matical extension. 

The  importance  of  this  distinction  and  discussion  will  appear 
when  we  come  to  consider  the  question  of  genus  and  species, 
or  essentia  and  differentia.  At  present  it  suffices  merely  to 
specify  the  qualifications  under  which  the  law  is  true. 

2.  The  second  limitation  of  the  formula  regarding  the  rela- 
tion between  extension  and  intension  is  that  it  is  not  applicable 
to  all  conceptions  independently  of  the  relation  between  genus 
and  species.  Thus  the  extension  of  "  man "  cannot  be  com- 
pared with  that  of  "bear  ; "  of  "horse  "  with  "lion  ;  "  of  "gov- 
ernment" with  "science."  Nor  can  any  measure  of  the  rela- 
tion between  the  extension  and  the  intension  of  such  terms  be 
determined.  Hence  the  law  cannot  apply  to  terms  taken  pro- 
miscuously. It  can  apply  only  to  conceptions  which  represent 
a  superordinate  and  a  subordinate  notion,  or  a  genus  and  a 
species.  Neither  genera  nor  species  can  be  compared  in  this 
relation  with  their  own  kind,  but  only  with  each  other.  The 
comparision  must  be  limited  to  terms  representing  different 
degrees  of  generalization.  All  concepts  may  thus  be  brought 
under  one  comprehensive  head,  but  apart  from  such  a  relation 
their  extension  and  intension  will  not  be  an  object  of  deter- 
mination at  all.  Hence  this  limitation  of  the  law.  But  it 
should  be  remarked  that  this  does  not  set  aside  the  formula. 
It  only  qualifies  it  and  its  application.  In  its  proper  concep- 
tion and  under  appropriate  conditions  it  remains  valid,  al- 
though it  may  be  possible  to  exaggerate  its  importance. 


INTENSION  AND  EXTENSION  OF  CONCEPTS        79 

But  even  those  who  criticise  the  doctrine  admit  considerable 
importance  for  the  meaning  which  they  assume  is  latent  at  the 
basis  of  it.  Bosanquet,  who  subjects  the  formula  to  a  some- 
what searching  criticism,  virtually  admits  all  that  the  law  ever 
meant  to  express  by  saying  that  conceptions  may  vary  their 
import  and  range  of  application  in  a  way  much  as  the  doctrine 
of  an  inverse  ratio  asserts.  He  says:  "It  is  certain  that  to 
abstract  and  to  distinguish — to  know  what  belongs  to  one  re- 
lation, and  what,  again,  though  conjoined  with  that  relation, 
yet  does  not  arise  out  of  it,  but  out  of  some  other  condition 
or  caiise — is  the  first  duty  of  the  scientific  intelligence.  In 
consequence  of  this  activity  arrangements  of  individual  objects 
under  a  series  of  abstractions,  each  applying  to  a  wider  aggre- 
gate than  the  last,  meet  us  on  every  hand  and  most  obviously 
of  all  in  family  relationships  as  estimated  among  civilized  na- 
tions."* When  we  take  into  account,  therefore,  that  the 
diminution  of  intension,  in  any  case,  does  not  involve,  neces- 
sarily, a  decrease  in  the  number  of  qualities  in  the  individual 
or  sub-class,  but  only  the  number  of  common  qualities  consti- 
tuting the  genus,  as  before  determined,  and  so  with  the 
increase  of  the  same,  there  will  be  nothing  to  seriously  object 
to  in  the  law,  except  its  precise  mathematical  and  promiscuous 
application.     Its  importance  is  therefore  vindicated. 

3d.  The  Denotation  and  Connotation  of  Terms. — 
Some  logicians  use  this  distinction  as  identical  with  that  be- 
tween the  extension  and  the  intension  of  terms.  Others  em- 
ploy it  in  a  somewhat  different  sense.  Indeed,  it  is  only  this 
variation  of  usage  and  its  frequent  coincidence,  or  close  con- 
nection with  the  application  of  extension  and  intension,  that 
makes  it  necessary  to  consider  the  matter  at  all.  I  shall  notice 
briefly  the  doctrines  of  Mill,  Fowler,  and  Keynes. 

Mill's  distinction  is  between  connotative  and  non-connotative 
terms.  "  A  non-connotative  term,"  he  says,  "is  one  which  signi- 
fies a  subject  only,  or  an  attribute  only.  A  connotative  term  is 
one  which  denotes  a  subject  and  implies  an  attribute.  By  a  sub- 
ject is  here  meant  anything  which  possesses  attributes.  Thus 
*  See  Bosanquet's  Logic,  Vol.  I.,  Introduction,  §  8.  pp.  46-71. 


80  ELEMENTS  OF  LOGIC 

John,  or  London,  or  England,  are  names  which  signify  a  sub- 
ject only.  Whiteness,  length,  virtue,  signify  an  attribute  only. 
None  of  these  terms,  therefore,  are  connotative.  But  white, 
long,  virtuous,  are  connotative."  The  distinction  here  made 
Mill  professes  to  be  a  restoration  of  scholastic  usage.  The  in- 
ference is,  as  he  himself  expresses  it,  that  "  all  concrete  gen- 
eral names  are  connotative  ; "  also  adjectives  and  "  proper 
nouns  are  not  connotative."  As  far  as  can  be  determined 
from  his  language  "  denotative  "  merely  means  "  significa- 
tive ; "  for  he  does  not  define  it.  He  practically  identifies  it 
with  "  non-connotative."  "  Connotative  "  he  defines  accord- 
ing to  scholastic  usage  as  equal  to  "  connotare,  to  mark  along 
with  ;  to  mark  one  thing  ivith,  or  in  addition  to  another." 
This  identifies  it  with  imply,  or  the  implicative  power  of  a 
term. 

Objections  can  be  produced  with  considerable  force  to  his 
classification  of  connotative  and  non-connotative  terms  on  the 
basis  of  these  definitions.  For  instance,  he  says  that  proper 
names  "  do  not  indicate  or  imply  any  attributes  as  belonging 
to  individuals."  This  can  be  directly  challenged.  Bucephalus, 
The  Secretary  of  State,  Mont  Blanc,  quite  distinctly  imply  at- 
tributes. It  does  not  suffice  to  make  a  proper  name  that  we 
merely  capitalize  a  word.  It  is  such  only  by  virtue  of  its  appli- 
cation to  a  single  individual.  But  if,  besides  denoting  a  subject 
it  implies  an  atti'ibute  or  attributes,  it  becomes  connotative. 
Again,  an  adjective  is  as  much  the  name  of  an  attribute  as  an 
abstract  noun  formed  from  it,  and  it  quite  as  distinctly  im- 
plies something  else.  It  is  true  that  it  does  not  denote  a  sub" 
ject  and  imply  an  attribute,  but  it  denotes  an  attribute  and 
implies  a  subject,  and  as  connotative  is  regarded  by  Mill  as 
synonymous  with  implication,  adjectives  must  be  quite  as  con- 
notative as  their  abstract  substantives.  Indeed,  the  implicating 
power  of  abstract  substantives  is,  if  anything,  less  than  their 
original  adjectives,  because  they  are  conceived  as  substantives 
or  subjects,  whose  relation  is  less  apparent  to  consciousness 
than  attributives.  Hence  if  either  of  the  two  are  non-connota- 
tive, it  should  be  the  abstract  nouns. 


INTENSION  AND  EXTENSION  OF  CONCEPTB        8 J 

It  will  be  apparent  in  Mill's  usage  that  the  terms  cannot  be 
identical  with  that  of  extension  and  intension,  and  hence  his 
doctrine  either  has  a  diminished  importance  or  is  too  confus- 
ing in  its  implications  to  be  of  much  service  in  the  problem 
under  consideration.  Fowler  and  Keynes  take  the  liberty  to 
modify  their  application  or  meaning  so  as  to  coincide  with 
extension  and  intension,  "  denotation  "  being  identical  with  ex- 
tension and  "  connotation  "  with  intension.  In  this  applica- 
tion they  require  no  further  discussion.  If  I  might  be  allowed 
to  introduce  an  innovation,  I  would  prefer,  since  we  already 
have  the  terms  extension  and  intension,  to  employ  "denota- 
tive "  to  indicate  the  application  of  a  name  to  individuals  with- 
out any  reference  to  their  qualities,  and  hence  only  to  denote 
them  as  concrete  wholes,  numerically  or  mathematically  con- 
sidered. "  Connotative  "  I  would  employ  to  connote  or  com- 
prehend a  union  of  attributes,  or  a  class  of  individuals.  It 
would  thus  apply  to  individual  and  general  wholes,  one  as  an 
aggregate  of  qualities  and  the  other  as  an  aggregate  of  indi- 
vidual objects.  Denotative  might  apply  to  the  names  of  singu- 
lar attributes  or  singular  individuals  as  such.  In  this  way  I 
should  remain  by  the  strict  import  of  the  word  connote,  and 
gain  an  economical  term  for  a  logical  synthesis  of  any  kind, 
and  retain  a  separate  one  for  all  concepts  representing  an  in- 
dividual unit  of  any  kind.  In  this  sense  I  have  used  the  terms 
where  I  found  it  necessary  to  employ  them  technically  at  all. 
But  as  no  particular  logical  doctrine  is  dependent  upon  tbis 
usage,  I  do  not  urge  either  the  adoption  or  the  importance  of 
it.* 

*  For  general  discussion  of  this  question  and  the  problem  relating  to 
the  extension  and  intension  of  terms  the  student  may  consult  the  follow 
ing  references  :  Mill  :  Logic,  Book  I.,  Chap.  II.,  §  5  ;  Fowler  :  Deductive 
Logic,  Chap.  II.;  Keynes:  Formal  Logic,  Chap.  II.  ;  Hamilton:  Lectures 
on  Logic,  Lects.  XI.  and  XII.  ;  Venn:  Empirical  Logic,  Chap.  VII.  ; 
Bosanquet :  Logic,  Vol.  I.,  Introduction,  §  8,  pp.  46-71;  De  Morgan: 
Formal  Logic,  Chap.  XII. 

6 


CHAPTER  VI. 

DEFINITION    AND    DIVISION 

1st.  The  Predicables. — Definition  and  Division  are  com- 
plicated processes,  the  former  especially,  since  there  are  several 
kinds  of  definition.  But  to  understand  them  and  their  place 
in  logical  science  and  disquisition  we  must  examine  into  the 
nature  and  meaning  of  the  so-called  predicables.  They  are 
usually  stated  as  five  in  number,  as  follows  : 

Genus         (yevos)  =  Genus. 

Species        (eiSos)  =  Species. 

Differentia  (8ia<popa)       =  Difference. 
Propriuin  (I8i6v)  —  Property. 

Accidens     (a-vfj.Pefir]K6s)  =  Accident. 

The  most  natural  order  of  considering  the  predicables  would 
be  that  in  which  they  are  stated.  But  as  the  meaning  of  the 
first  three  is  dependent  upon  understanding  the  fourth,  and 
perhaps  the  fifth,  I  shall  begin  with  the  fourth  and  return  to 
the  others. 

1.  Property. — This  term  is,  for  all  practical  purposes,  synony- 
mous with  quality  and  attribute.  Some  writers  endeavor  to 
distinguish  between  them,  but  the  distinction  serves  no  im- 
portant logical  purpose.  Hence  by  a  property  of  a  thing  I 
mean  any  quality,  mark,  characteristic,  or  attribute  of  it  which 
goes  to  make  it  what  it  is.  Thus  whiteness  is  a  property  of 
snow  ;  hardness,  of  iron  ;  yellowness,  of  gold  ;  brilliancy,  of  a 
diamond  ;  instability,  of  a  liquid,  etc.  It  is  the  same  with  any 
namable  quality  of  an  object.  But  qualities  or  properties  are 
not  all  of  the  same  value  or  importance  to  the  existence  of  a 
thing.  Hence  they  are  usually  divided  into  essentia/  find  non- 
essential. Essential  properties  are  those  which  are  necessary 
to  the  particular  nature  of  an  object,  which  would  not  be  what 


DEFINITION  AND  DIVISION  83 

it  is  except  for  these  qualities.  For  example,  the  essential 
property  of  a  pen  is  that  it  be  fit  for  writing  ;  of  man,  that  he 
have  life  and  consciousness  ;  of  a  lamp,  that  it  be  able  to  give 
light ;  of  a  tree,  that  it  be  of  wood,  have  a  trunk  and  branches, 
etc.  ;  of  iron,  that  it  have  a  certain  density,  metallic  lustre,  and 
molecular  cohesion.  In  a  great  many  cases  it  is  difficult  to 
assure  ourselves  that  a  given  quality  is  essential  rather  than 
non-essential.  This  is  because  of  the  indeterminate  extension 
of  the  term  under  notice.  Thus  we  might  consider  sweetness 
as  an  essential  property  of  sugar,  and  so  it  is  in  the  common 
use  of  the  term.  But  to  the  student  of  chemistry  this  is  not 
necessarily  the  case,  as  we  have  seen  that  science  will  often 
generalize  a  term  without  carrying  along  with  the  increased 
extension  the  essential  property  of  its  narrower  import.  An 
instance  of  this  is  the  term  "  metal."  Originally  it  was  sup- 
posed that  a  specific  gravity  greater  than  water  was  a  neces- 
sary property  of  metals.  But  on  the  discovery  that  potassium, 
sodium,  and  lithium  were  metals  because  of  their  metallic 
lustre  and  structure,  this  property  became  one  of  their  essen- 
tial characters,  and  their  specific  gravity  was  disregarded  by  the 
term  metal  obtaining  a  more  general  application.  We  have, 
therefore,  always  first  to  determine  the  extension  of  a  concept 
before  indicating  its  essential  properties.  Fluidity  will  be  an 
essential  property  of  water,  if  we  do  not  include  the  term  ice 
in  it.  Animal  fibre  is  an  essential  property  of  "  meat,"  if  we 
do  not  use  the  term  synonymously  with  the  term  "  food."  But 
all  this  merely  indicates  how  difficult  it  is  to  name  absolutely 
the  essential  property  denoted  by  a  given  term.  We  have  first 
to  settle  what  extension  is  given  it,  and  this  will  be  determined 
by  the  limits  assigned  to  the  presence  of  a  given  quality,  and 
this  quality  will  be  an  essential  one  so  far  as  it  is  identified 
with  the  extension  of  the  term,  or  made  to  determiue  that  ex- 
tension. This  last  modifying  clause  has  to  be  added  because 
some  universal  properties  are  regarded  as  non-essential.  But 
the  essential  properties  will  determine  the  limits  of  extension 
for  any  given  meaning,  but  will  not  stand  in  the  way  of  a 
higher  generalization  eliminating  that  quality  as  essential,  and 


84  ELEMENTS  OF  LOGIC 

substituting  another  in  its  place.  But  in  this  case  the  concept 
is  materially  changed,  and  is  not  the  same  as  before.  In  par- 
ticular cases,  therefore,  before  naming  the  essential  quality  we 
must  see  that  the  term  is  clearly  understood  or  that  its  exten- 
sion is  not  greater  and  cannot  be  greater  than  the  property  to 
be  specified. 

A  non-essential  property  is  one  which  is  not  necessary  to  the 
concept  or  existence  of  a  thing.  For  instance,  whiteness  is 
not  an  essential  property  of  man  ;  redness,  of  an  apple  ;  a 
specific  length,  of  the  sides  of  a  triangle ;  iron  material,  of  a 
ship,  etc.  The  non-essential  properties  are  called  accidents, 
which  are  admirably  defined  and  illustrated  by  Jevons.  "  An 
accident,"  he  says,  "is  any  quality  which  may  indifferently  be- 
long or  not  belong  to  a  class,  as  the  case  may  be,  without 
affecting  the  other  qualities  of  the  class.  The  word  means 
that  which  falls  or  happens  by  chance,  and  has  no  necessary 
connection  with  the  nature  of  the  thing.  Thus  the  absolute 
size  of  a  triangle  is  a  pure  accident  as  regards  its  geometrical 
properties  ;  for  whether  the  side  of  a  triangle  be  one-tenth  of 
an  inch  or  a  million  miles,  whatever  Euclid  proves  to  be  true 
of  one  is  true  of  the  other.  The  birthplace  of  a  man  is  an 
accident  concerning  him,  as  are  also  the  clothes  in  which  he  is 
dressed,  the  position  in  which  he  rests,  and  so  on.  Some 
writers  distinguish  between  separable  and  inseparable  acci- 
dents. Thus  the  clothes  in  which  a  man  is  dressed  is  a  separ- 
able accident,  because  they  can  be  changed,  as  can  also  his 
position,  and  many  other  circumstances  ;  but  his  birthplace, 
his  height,  his  Christian  name,  etc.,  are  inseparable  accidents, 
because  they  can  never  be  changed,  although  they  have  no 
necessary  or  important  relation  to  his  general  character." 

Accidental  properties  are  almost  as  indeterminate  in  par- 
ticular cases  as  the  essential.  This  is  because  the  term  may, 
like  that,  be  relative  in  its  import.  That  is,  what  is  accidental 
in  one  relation  may  be  essential  in  another.  Thus  muddiness 
may  be  an  accidental  quality  of  rivers,  but  an  essential  quality 
of  a  certain  river.  A  better  illustration  is  that  of  red  hair.  It 
is  a  purely  accidental  property  of  a  man,  as  a  man,  but  it  may 


DEFINITION  AND  DIVISION  85 

be  essential  to  him  as  an  individual.  There  are  instances, 
however,  where  the  accident  will  hardly  be  considered  as  in 
any  way  essential  even  to  the  individual.  But  this  may  be 
controverted,  and  it  is  not  necessary  to  sustain  the  position 
for  our  purposes.  All  we  require  to  remember  is  that,  usually 
at  least,  the  accidents  are  relative  to  certain  qualities  regarded 
as  essential,  and  that  by  narrowing  the  extension  of  a  concept, 
we  may  make  them  essential  in  that  sphere.  The  only  place 
where  an  accident  will  be  absolutely  such  is  in  the  case  of  the 
individual  or  singular  concept,  and  even  here  it  may  be  dis- 
puted whether  the  distinction  between  essence  and  accident 
can  be  drawn.  It  may  require  for  its  existence  a  comparison 
with  some  other  individual  having  common  attributes  and 
certain  differences,  in  which  case  the  common  attributes  would 
be  the  essentia  of  a  common  name  and  the  difference  would 
be  the  accidents  of  it. 

If  this  be  the  case,  however,  it  only  sbows  the  ambiguity  of 
the  term  "accident."  In  fact,  it  has  more  than  one  significa- 
tion. Sometimes  it  is  used  synonymously  with  any  quality  of 
an  object,  and  in  this  case  it  is  equivalent  to  property  or  attri- 
bute. Again,  it  is  used  synonymously  with  differentia,  which 
is  the  property  defining  a  species.  In  this  sense  it  is  acci- 
dental in  one  relation  and  essential  in  another.  A  third  mean- 
ing is  that  which  denotes  a  property  in  no  way  essential  to 
either  the  genus,  the  species,  or  the  individual.  This  is  per- 
haps the  sense  in  which  it  is  usually  taken  by  logicians.  But 
it  is  sometimes  relative  in  this  usage.  Thus  seven  feet  stature 
might  be  an  accident  of  a  giant  as  a  man,  but  necessary  to  his 
being  a  giant.  But  a  case  where  this  relative  import  is  riot 
apparent,  if  it  exist  at  all,  would  be  the  size  of  the  hand,  the 
presence  of  a  mole  on  the  skin,  liability  to  blush,  etc.,  which 
would  be  regarded  as  accidents  of  the  individual.  This  is  the 
proper  sense  of  the  term,  although  it  may  be  disputed  whether 
it  is  ever  absolute  in  its  meaning  even  under  the  circum- 
stances just  considered.  But  it  is  not  necessary  to  decide 
this  point.  Its  relative  import  suffices  to  set  aside  certain 
properties   which    may   be   disregarded    in    determining   the 


86  ELEMENTS  OF  LOGIC 

essentially  important  meaning  of  a  concept.  It  only  remains 
to  remark  that  some  accidentice  are  regarded  as  universal  or 
coincident  with  the  genus  or  species,  as  curly  hair  of  the 
negro,  and  others  are  merely  casual  or  contingent,  as  sickness, 
flatness  of  the  nose,  shape  of  the  head,  etc. 

Some  logicians  use  the  distinction  peculiar  property  to  de- 
note what  belongs  to  a  whole  class  and  to  that  class  only,  as 
risibility  in  man.  But  this  comes  under  the  general  head  of 
universal  accidents,  as  one  kind  of  them,  and  needs  no  further 
consideration. 

2.  Differentia. — Differentia,  or  difference,  is  the  name  ol 
that  particular  property  which  distinguishes  one  species  from 
another.  For  example,  bi2)edality,  or  two-footedness,  is  a  dif- 
ferentia of  man  as  compared  with  quadrupeds  ;  the  possession 
of  feathers,  a  differentia  of  birds  as  compared  with  horses ; 
cloven-footedness  of  cattle  is  a  differentia  as  compared  with 
horses  ;  redness  of  core  or  pulp  is  a  differentia  of  "  blood- 
oranges  "  as  compared  with  ordinary  oranges,  etc.  In  all  such 
cases  the  difference,  or  differentia,  is  a  quality  or  property  in 
addition  to  the  generic  or  common  qualities,  and  merely  de- 
termines the  species  or  individual  under  the  class  or  genus.  In 
this  respect  or  relation  it  may  even  be  spoken  of  as  essential, 
although  it  is  essential  only  to  the  species  or  individual. 

There  is  no  word  used  by  logicians  as  the  counterpart  or 
complement  of  differentia  except  the  term  genus.  But  this  is 
also  contrasted  with  species,  and  species  is  the  name  for  an 
individual  or  narrower  class  rather  than  of  a  quality  or  group 
of  qualities  expressed  by  the  term  differentia,  although  it  is  de- 
termined by  this  characteristic.  Sometimes  the  term  essence, 
however,  is  used  as  the  opposite  of  difference,  and  denotes  the 
quality  or  qualities  essential  or  necessary  to  the  existence  of 
a  class  or  individual.  It  is  identical  with  essential  property. 
For  example,  vertebrate  anatomy  and  rationality  are  essential 
to  the  class  "  man  ; "  four-footedness  to  the  class  quadrupeds  ; 
cloven-footedness  to  cattle  ;  a  certain  pungent  taste,  color, 
etc.,  to  oranges ;  woody  structure,  trunk,  and  branches  to 
trees,  etc.     But  here,  as  in  the  case  of  the  term  differentia,  it 


DEFINITION  AND  DIVISION  87 

is  relative  in  its  import.  Besides,  it  is  somewhat  ambiguous 
in  that  it  is  sometimes  used  to  denote  the  common  qualities 
of  the  class  or  genus,  and  at  other  times  is  applicable  to  the 
differentia  as  an  essential  quality  of  the  species.  Usage  is  not 
so  uniform  as  is  desirable  in  this  matter.  Indeed,  Whately 
calls  the  differentia  the  formal  essence  of  a  thing,  and  the 
genus  the  material  part  or  essence  of  a  concept. 

The  confusion  incident  to  the  discussion  may  be  avoided  if 
we  adopt  a  new  term  corresponding  to  differentia,  as  genus 
corresponds  to  species,  and  retain  essence  or  essentia  as  con- 
trasted only  with  accident  or  accidentia.  There  has  been 
some  difficulty  in  selecting  a  proper  term  that  would  express 
the  characteristic  meant  to  be  indicated  by  the  term.  But 
the  best  I  have  been  able  to  accomplish  in  this  matter  is  the 
selection  of  conferentia  (from  con  and  fero,  to  bring  together, 
as  differentia  is  from  dis  and  fero,  to  separate).  I  had  seri- 
ously thought  of  using  communia  (neuter  plural  of  the  Latin 
commune,  what  is  common),  because  it  would  etymologically 
import  the  common  qualities.  But  the  fatal  objection  was 
that  no  singular  of  it  could  be  used  without  confusion  with 
the  English  commune.  I  have  therefore  chosen  "conferentia," 
as  a  good  logical  and  etymological  complement  of  differentia. 
It  has  the  objection  to  contend  with  that  the  English  equiva- 
lent, "  conference,"  is  not  associated  with  any  logical  usage  of 
the  kind  here  wanted.  But  we  can  confine  ourselves  to  the 
form  "conferentia,"  with  the  distinctive  meaning  assigned 
to  it. 

By  "  conferentia "  I  shall  mean  the  common  quality  or 
qualities  which  "  bring  together  "  like  individuals,  or  consti- 
tute the  application  of  a  term  to  a  class.  It  is  therefore  the 
essence  of  the  genus,  as  the  differentia  is  the  essence  of  the 
species.  It  is  also  a  relative  term,  as  what  is  conferentia  in 
one  class  may  be  the  differentia  in  relation  to  a  co-ordinate 
species.  But  in  it  we  have  a  convenient  expression  for  the 
common  qualities  uniting  individuals  iu  a  class,  in  contrast 
with  those  important  qualities  that  are  not  common  to  all 
under  it.     Generic  and  general,  as  well  as  "  conferentia!,"  may 


8S  ELEMENTS  OF  LOGIC 

be  the  corresponding  adjectives.  In  this  way  we  may  reserve 
essence,  or  essentia,  to  denote  the  essential  properties  of  either 
genus  or  species,  and  accidentia  for  the  non-essential.  The 
following  table  summarizes  the  discussion  and  classifies  prop- 
erties : 

{  i  Conferent.ia=  Common  properties,  or  essence  of  the  Genus. 

p  J  Essentia     ■<  Differentia  =  Distinctive  properties,  or  essence  of  the  Spe- 

ert P"  1  '         cies- 

Accidentia  ]  p        ,        |  Non-essential  to  either  Genus  or  Species. 

4.  Genus  and  Species. — These  are  important  terms  to  define. 
The  concepts  to  which  they  apply  play  a  very  large  part  in  all 
logical  discourse.  They  are  closely  related  to  the  extension 
and  the  intension  of  concepts.  They  differ  from  them  only  in 
implying  a  distinction  between  the  general  or  conferential  and 
differential  properties  of  concepts,  which  is  not  necessarily  in- 
volved in  the  difference  between  extension  and  intension,  al- 
though when  the  variation  between  these  is  in  an  inverse  ratio 
it  coincides  with,  and  perhaps  in  a  measure  determines,  the  dis- 
tinction between  genus  and  species.  But  this  relation  is  a  matter 
for  more  advanced  Logic  to  consider.  The  question  of  import- 
ance in  all  practical  reasoning  is  the  influence  exerted  upon  it 
by  those  various  conceptions  known  as  genera  and  species,  and 
the  logical  relation  between  them.  It  will  be  seen  most  clearly 
in  the  fallacies  of  Equivocation  and  Accident,  when  we  come  to 
consider  them.  It  will  also  be  valuable  in  interpreting  the  rela- 
tion between  subject  and  predicate  in  one  class  of  judgments, 
the  process  of  Conversion,  and  the  relation  subsisting  between 
essential  and  accidental  properties,  on  the  one  hand,  and  con- 
ferential and  differential  properties,  as  concomitants,  on  the 
other.  We  now  proceed  to  define  and  illustrate  the  two  terms 
very  carefully. 

A  conception  which  applies  to  a  whole  class  of  objects  is 
called  a  genus.  Thus  "  man  "  is  a  genus-concept  because  it 
is  a  name  which  applies  to  the  various  kinds  of  individuals, 
tribes,  or  nations  of  men.  "  Substance  "  is  a  genus  because  it 
includes  iron,  clay,  brass,  gold,  silver,  water,  etc.  It  is  a 
general  name  for  various  species  and  individuals  under  it. 


DEFINITION  AND  DIVISION  89 

A  conception  which  applies  to  a  narrower  class,  or  an  indi- 
vidual under  a  genus,  is  a  species.  Thus  "  Caucasian  "  is  a 
species  compared  with  the  genus  man  ;  "  iron  "  is  a  species 
compared  with  the  genus  substance  ;  triangle  is  a  species  com- 
pared with  the  genus  figure.  A  species,  therefore,  has  the  less, 
and  a  genus  the  greater,  extension. 

But  it  must  be  farther  observed  that  the  terms  "  Caucasian  " 
and  "  iron  "  may  also  be  genera  in  relation  to  a  lower  order  of 
concepts.  Thus  "  Caucasian  "  includes  Germans,  Frenchmen, 
Englishmen,  etc.  ;  iron  includes  steel,  malleable  iron,  wrought- 
iron,  cast-iron,  etc.  On  the  other  hand,  "  man  "  is  a  species 
in  comparison  with  the  higher  orders  "  biped,"  "  vertebrate," 
"  organic  being,"  etc.  To  take  a  single  term  illustrating  both 
relations,  the  concept  "  metal "  is  a  genus  compared  with 
"iron,"  "gold,"  "  silver,"  "platinum,"  etc.,  but  a  species  com- 
pared with  the  term  "  substance." 

From  these  illustrations  it  is  apparent  that  genus  and  spe- 
cies are  relative  terms  wherever  they  are  convertibly  applicable 
to  the  same  concept  in  different  relations.  It  will  be  noticed 
that  a  term  is  always  a  genus  in  relation  to  a  narrower  exten- 
sion, and  a  species  in  relation  to  a  wider  extension.  We  may 
thus  proceed  in  either  direction  until  we  reach  the  limits  of 
farther  progress,  as  Being,  organized  being,  vertebrate,  man, 
American,  Lincoln.  Here  all  intermediate  terms  are  either 
genera  or  species  according  as  they  are  conceived  in  relation 
to  a  higher  or  a  lower  order ;  according  as  they  include  a 
lower  class,  or  are  included  in  a  higher.  But  the  two  extreme 
terms  cannot  be  viewed  in  this  twofold  relation.  "  Being  " 
is  a  genus,  but  not  a  species.  On  the  other  hand,  the  term 
" Lincoln"  is  a  species,  but  not  a  genus,  supposing,  of  course, 
that  we  can  speak  of  an  individual  as  a  species.  The  former 
is  not  a  species,  because  there  is  no  higher  genus  under  which 
objects  can  be  brought,  and  the  latter  is  not  a  genus  because 
it  cannot  be  divided  into  individuals  or  species.  All  singular 
terms  are  species,  but  not  genera.  They  are  called  infima 
species,  or  lowest  species.  They  are  always  individuals,  so  far 
as  Logic  is  concerned.     On  the  other  hand,  the  highest  genus 


90  ELEMENTS  OF  LOGIC 

is  called  the  summum  genus,  or  genus  generalissimum.  It  is 
represented  by  some  such  term  as  "being,"  "thing,"  "some- 
thing," "  ultimate  reality,"  but  in  all  cases  must  be  repre- 
sented by  a  single  concept.  It  is  thus  worthy  of  remark  that 
there  is  only  one  absolute  summum  genus,  while  there  may 
be  an  indefinite  number  of  infima  species.  All  intermediate 
concepts  are  sometimes  called  subalterns,  as  being  either  gen- 
era or  species,  according  to  the  relation  in  which  they  are 
viewed. 

It  is  necessary  to  notice  the  use  of  the  terms  genus  and  spe- 
cies as  used  in  natural  history.  A  species  is  there  "  a  class  of 
plants  or  animals  supposed  to  have  descended  from  common 
parents,  and  to  be  the  narrowest  class  possessing  a  fixed  form  ; 
a  genus  is  the  next  higher  class."  This  is  Jevons's  definition, 
but  he  does  not  illustrate  it.  Perhaps  in  natural  history  the 
term  "  tree  "  would  represent  a  genus,  and  oak,  elm,  maple, 
etc.,  a  species,  while  the  distinct  kinds  of  oak,  elm,  and 
maple  would  be  varieties.  But  the  peculiar  use  of  the  term 
species  here  is  that  it  is  supposed  to  be  fixed,  and  not  relative 
as  in  Logic.  In  this  conception  of  the  term  its  meaning  is 
quite  distinct  from  that  of  Logic,  where,  as  we  have  seen,  any 
but  the  summum  genus  and  the  infima  species  may  be  either  a 
genus  or  species,  according  to  its  relation  to  a  higher  or  lower 
order.  In  natural  history  it  is  supposed  to  represent  certain 
fixed  characters  and  relations  to  a  common  progenitor.  But 
the  acceptance  of  the  doctrine  of  evolution  prevents  any  such 
determinate  line  of  distinction  from  being  drawn.  The  con- 
ception of  "  species  "  becomes  an  indistinct  one,  and  does  not 
imply  any  necessary  assumption  about  a  particular  common 
ancestor.  It  denotes  only  a  certain  aggregate  of  characteris- 
tics with  differences  less  marked  and  distinct  than  between 
genera.  This  change  of  meaning,  therefore,  makes  the  term 
approximate  its  logical  import,  if  it  may  not  ultimately  iden- 
tify the  two.  But  it  is  necessary  to  remark  the  differences  that 
have  hitherto  prevailed  between  the  logical  use  of  the  term 
and  its  use  in  natural  history. 

It  is  important  to  notice  an  ambiguity  in  the  logical  use  of 


DEFINITION  AND  DIVISION  91 

the  term  genus.  This  ambiguity  is  apparent,  as  I  have  already 
remarked  in  the  contrast  between  genus  and  species,  on  the 
one  hand,  and  the  contrast  between  genus  and  differentia,  on 
the  other,  where  species  and  differentia  are  not  identical.  The 
fact  determines  a  double  use  of  the  term  genus.  Differentia, 
as  we  have  seen,  denotes  certain  properties  determining  the 
species,  but  it  does  not  determine  the  whole  content  or  inten- 
sion of  the  species.  It  denotes  only  the  distinctive  qualities. 
The  species  is  therefore  a  concrete  thing,  combining  the  dif- 
ferential and  conferential  qualities,  or  the  common  qualities 
expressed  or  implied  by  the  "  genus  "  and  the  added  differen- 
tial quality  which  makes  it  a  species  distinct  from  some  other 
species  having  less  or  different  qualities.  But  strictly  taken, 
a  specific  term  is  concrete  and  denotes  indifferently  an  indirid- 
aal  or  an  aggregate  of  qualities.  It  implies  no  special  relation 
between  extension  and  intension.  But  the  genus,  or  terms  that 
represent  genera,  are  not  so  distinct  in  their  meaning.  On 
the  one  hand,  they  denote  numerically,  that  is,  extensively, 
all  that  is  denoted  by  the  various  species  under  them  ;  logi- 
cally, or  intensively,  they  denote  less  than  any  given  species. 
In  relation  to  species,  therefore,  genera  are  greater  in  exten- 
sion and  less  in  intension,  but  in  so  far  as  they  denote  the 
common  qualities  they  apply  equally  to  individual  species  and 
to  the  whole  number  of  species,  while  specific  terms  will  not 
apply  to  the  whole  of  the  genus.  The  relation  between  genus 
and  species  in  this  respect  can  be  illustrated  as  follows : 
"  Man,"  as  a  genus,  applies  equally  to  Germans,  Frenchmen, 
Englishmen,  or  Caucasians,  Mongolians,  Negroes,  etc.  The 
only  difference  between  them  is  quantity  or  quality  of  inten- 
sion. Germans  contain  certain  qualities  which  Frenchmen  do 
not.  But  the  difference  is  not  great  enough  to  prevent  the 
use  of  the  term  "  man  "  to  denote  both  species.  In  this  sense 
the  genus,  or  man,  is  identical  with  all  the  species  taken  together 
and  extensively  considered.  The  contrast,  therefore,  between 
genus  and  species  is  not  a  complete  one.  Indeed,  strictly 
speaking,  they  cannot  be  contrasted  at  all.  They  differ  only 
in  regard  to  the  number  of  individuals  denoted  by  them,  and 


92  ELEMENTS  OF  LOGIO 

in  this  respect  are  alike  concrete  in  their  signification.  As 
applying  to  all  the  species  the  genus  is  but  a  common  name 
for  a  number  of  individual-wholes,  and  applies  mathematically 
to  all  alike.  The  fact  that  genus  and  species  are  not  wholly 
distinct  is  apparent  in  all  common  judgments  involving  purely 
class  terms.  Thus,  "Man  is  a  biped,"  or  "All  Germans  are 
men."  I  cannot  reverse  the  subject  and  predicate  because  the 
difference  of  extension  between  them  will  not  permit  of  it. 
There  is,  however,  a  connection  always  subsisting  between 
genus  and  species  which  allows  a  statement  in  the  form  of  a 
simple  proposition.  But  no  such  connection  exists  between 
the  different  species  under  the  same  genus.  Thus  I  can  never 
say  "  the  Germans  are  Frenchmen."  Of  them  I  can  only  form 
a  negative  proposition,  where  no  relation  of  extension  exists. 
The  distinction,  therefore,  between  species  is  absolute.  Be- 
tween genus  and  species  it  is  not  absolute,  but  is  only  of  quan- 
tity, ivhether  of  extension  or  intension. 

But  when  the  term  genus  is  contrasted  with  the  term  dif- 
ferentia the  matter  is  quite  different.  Genus  and  differentia 
are  contrasted  not  as  larger  and  smaller  classes  of  individual- 
wholes,  but  as  different  qualities  or  groups  of  qualities.  In 
this  sense  genus,  or  a  generic  concept,  denotes  or  connotes 
certain  common  qualities  which  characterize  the  whole  class, 
but  which  are  quite  distinct  from  the  differential  qualities 
which  form  the  differential  essence  of  the  species.  In  this 
meaning  the  genus  is  as  distinct  from  the  differentia  as  one 
species  is  from  another.  It  cannot  apply  as  a  name  to  any  of 
the  qualities  rejn-esented  by  the  differential  elements.  It  is 
only  a  name  for  the  common  qualities  of  a  class.  Thus  "  man," 
as  a  genus  contrasted  with  differentia,  does  not  denote  individ- 
ual -wholes  at  all,  but  only  a  certain  group  of  common  qualities. 
It  is,  therefore,  not  only  an  abstract  conception,  thus  con- 
ceived, but  denotes  only  the  intension  of  the  concept,  exclusive 
of  the  differentia,  and  so  serves  as  the  basis  for  determining 
its  extension.  But  the  extension  is  not  the  matter  in  thought 
when  conceiving  it  as  contrasted  with  the  differentia.  It  de- 
notes or  connotes  only  the  common  qualities,  and  anything 


DEFINITION  AND  DIVISION  93 

affirmed  of  it  in  this  sense  will  not  agree  with  the  species  or 
differential  concept.  For  example,  I  may  say,  "Pine  wood  is 
good  for  lumber."  This  is  not  true  of  every  specific  form  of 
j)ine  wood,  but  only  of  its  substance  or  generic  qualities. 
Matches  might  be  made  of  pine  wood  and  yet  not  be  good  for 
lumber.  The  affirmation,  therefore,  can  be  true  only  of  the 
genus  as  contrasted  with  the  differentia,  and  not  of  the  genus 
including  all  the  species.  The  distinction,  therefore,  between 
the  two  kinds  of  genera,  as  here  drawn,  is  the  same  as  that  be- 
tween the  two  kinds  of  general  terms,  the  mathematical  and  the 
logical.  Hence  I  distinguish  the  mathematical  and  the  logical 
genus :  the  first,  or  mathematical,  to  denote  the  genus  as  ap- 
plicable numerically  to  all  the  species,  and  the  second,  or  logi- 
cal, to  denote  or  connote  simply  the  conferentia  or  common 
essence,  and  not  affirmable  of  specific  characters. 

The  importance  and  meaning  of  this  distinction  must  be 
brought  out  by  further  illustration.  To  effect  this  requires  a 
brief  explanation  of  what  is  meant  by  the  connection  between 
subject  and  predicate.  Usually  we  suppose,  or  are  told,  that  the 
predicate  is  more  or  less  identical  with  the  subject.  Thus  if  I 
say,  "Man  is  a  biped,"  I  mean  that  two-footedness  is  a  quality 
of  man.  But  I  may  mean  by  a  similar  judgment  that  one 
quality  invariably  and  universally  accompanies  another.  Thus 
to  say,  "Man  is  intelligent,"  may  mean  that  along  with  a  cer- 
tain representative  quality  or  qualities  standing  for  man  will 
be  found  the  equally  universal  and  necessary  quality  intelli- 
gence, but  which  is  not  analytically  represented  to  conscious- 
ness in  the  mention  of  the  name  "  man."  In  other  words,  I 
affirm  the  agreement  or  concomitance  of  certain  conferential 
qualities,  and  this  will  be  true  of  their  connection,  generically 
considered,  wherever  found.  To  take  the  former  illustration, 
"  Pine  wood  is  good  for  lumber,"  goodness  for  lumber  is  con- 
nected with  the  generic  qualities  of  pine  wood,  but  it  is  not 
connected  with  every  particular  or  specific  form  of  it.  The 
statement  undoubtedly  assumes  a  particular  form  and  quantity 
of  the  pine  wood  as  essential  to  its  making  lumber.  But  this 
only  shows  that  it  is  not  the  mathematical  genus  of  which  it  is 


94  ELEMENTS  OF  LOGIC 

affirmed,  but  only  of  the  logical,  and  hence  the  difficulty  when 
we  come  to  compare  the  predicate  with  the  species,  or  the  dif- 
ferential qualities. 

A  negative  illustration  will  bring  out  the  same  truth.  For 
instance,  I  cannot  say  that  "  All  men  are  white,"  but  I  can  say, 
"  The  Caucasians  are  white,"  because  "  whiteness  "  is  true  of 
the  species,  and  not  true  of  the  genus,  taken  either  logically 
or  mathematically.  But  now  if  I  turn  around  and  say,  "  The 
Caucasian  race  is  the  most  intelligent,"  this  may  not  be  true 
of  all  individual  Caucasians  numerically  considered,  but  only 
generally  ;  that  is,  their  intelligence  is  an  accidental  character- 
istic connected  with  such  essential  qualities  as  make  them 
men,  and  with  which  whiteness  is  found  frequently  enough  to 
make  the  statement  of  the  race  in  general.  We  therefore 
speak  of  the  logical  and  not  the  mathematical  genus  in  our 
proposition. 

The  importance  of  the  distinction  I  have  drawn  will  appear 
when  I  come  to  consider  the  doctrine  of  Judgments  and  Fal- 
lacies. But  it  is  fully  justified  in  the  ambiguity  remarked  in 
the  use  of  the  term  genus,  contrasting  it,  on  the  one  hand, 
with  the  species,  with  which  it  differs  only  quantitatively  in 
regard  to  extension,  and  on  the  other  with  the  differentia,  to 
which  it  is  opposed  and  with  which  it  differs  qualitatively  in 
regard  to  intension.  We  have,  therefore,  to  keep  constantly  in 
mind  that  all  generic  concepts  have  a  double  capacity,  the  math- 
ematical, and  the  philosophic  or  logical.  If  this  fact  is  closely 
guarded  the  student  will  be  saved  many  a  misstep  in  reasoning. 

2d.  Analysis  of  Concepts. — By  the  analysis  of  concepts 
I  mean  here  the  breaking  up  of  them  into  their  parts  and  sub- 
ordinate elements.  There  are  two  forms  of  analysis,  Division 
and  Partition,  which  must  be  considered  in  their  proper  order. 

1.  Division. — Division  is  the  analysis  of  the  extension  of  a 
concept,  the  separation  of  a  genus  into  its  species.  The  pro- 
cess is  usually  called  Logical  Division.  Thus  we  are  said  to 
divide  the  genus  "  tree  "  when  we  indicate  the  species  of  tree 
to  which  the  term  applies,  as,  for  example,  into  oak,  elm, 
maple,  willow,  ash,  pine,  etc.,  some  of  which  at  least  are  still 


DEFINITION  AND  DIVISION  95 

fai'ther  divisible  into  subordinate  species  or  individuals,  as 
oak  into  white,  black,  and  red  oaks  ;  pine  into  white  and  yel- 
low pines  ;  willow  into  white,  weeping,  and  swamp  willows, 
etc.  They  may  be  divided  by  some  other  principle  if  we  so 
desire.  Thus  we  might  divide  "tree  "  into  those  of  deciduous 
leaves,  and  those  of  evergreen  leaves,  etc.  It  is  not  necessary 
in  every  case  to  proceed  upon  the  same  principle.  But  what- 
ever principle  is  used  is  called  the  Fundamentum  Divisionis. 
In  order  to  be  a  principle  of  division  the  quality  or  circum- 
stance, taken  as  such,  "  must  be  present  with  some  and  absent 
from  others,  or  must  vary  with  the  different  species  compre- 
hended in  the  genus.  A  generic  property,  of  course,  being 
present  in  the  whole  of  the  genus,  cannot  serve  for  the  pur- 
pose of  division."  The  principle  of  division,  therefore,  must 
be  some  differentia,  or  differential  quality,  which  is  the  dis- 
tinctive feature  of  the  species.  Thus  if  I  divide  apples  into 
red,  green,  and  yellow,  color  is  the  principle  of  division,  and 
each  specific  color  is  the  differentia  of  its  class.  An  accidental 
property  will  not  suffice  for  any  permanent  or  scientific  division. 
Hamilton  enumerates  seven  rules  governing  the  process  of 
division.     They  are  : 

(a)  "  Every  division  should  be  governed  by  some  principle." 
(6)  "Every  division  should  be  governed  by  only  a  single 
principle." 

(c)  "The  principle  of  division  should  be  an  actual  and  essen- 
tial character  of  the  divided  notion,  and  the  division,  there- 
fore, neither  complex  nor  without  a  purpose." 

(d)  "  No  dividing  number  of  the  predicate  must  by  itself 
exhaust  the  subject." 

(e)  "The  dividing  numbers,  taken  together,  must  exhaust, 
but  only  exhaust,  the  subject." 

(f)  "  The  divisive  numbers  must  be  reciprocally  exclusive." 

(g)  "The  divisions  must  proceed  continuously  from  immedi- 
ate to  mediate  differences." 

These  are  not  all  of  equal  importance,  and  might  be  reduced 
to  a  smaller  number.  For  instance,  the  first  is  practically  the 
same  as  the  second.     The  fourth  and  fifth  (d  and  e)  might  be 


96 


ELEMENTS  OF  LOGIC 


summarized  in  one.  The  third  and  the  seventh  (c  and  g),  al- 
though important  in  a  complete  enumeration,  are  more  likely 
to  be  observed  naturally  than  some  of  the  others.  But  they 
require  to  be  kept  in  mind.  Jevons  reduces  them  to  three, 
which  serve  for  most  all  practical  purposes. 

The  importance  of  the  rules  is  seen  in  what  is  called  Cross 
Division,  which  is  the  naming  of  species  that  interpenetrate 
or  overlap.  Thus,  if  I  divided  trees  into  tall  trees,  green  trees, 
pine  trees,  and  dead  trees  ;  or  books  into  octavos,  histories, 
theoretical  books,  dictionaries,  etc.,  I  should  be  using  more 
than  one  principle  of  division,  and  indicating  species  that  were 
not  mutually  exclusive.  An  illustration  of  the  proper  form  of 
division  is  the  following  table,  or  outline  : 

f  Trilateral  =  Triangles. 

(  Parallelograms. 
Rectilinear  -j  Quadrilateral  =  •<  Trapezoids. 
(  Trapeziums. 
Plane  ■{  [  Multilateral  =  Polygons  of  more  than  four  sides, 

f  Circular  =  Circles. 
Curvilinear  J  Elliptic  =  Ellipse. 

I  Parabolic  =  Parabolas. 
Figures  \  {  Hyperbolic  =  Hyperbolas. 

(  Tetrahedrons. 

Rectilinear  ]  g^"*1™118- 

[  Parallelopipeds.  etc. 

{Spheres. 
Cylinders. 
Paraboloids. 

In  division  a  genus,  in  relation  to  a  species,  is  said  to  bo 
superordinate  ;  a  species  in  relation  to  a  genus  is  said  to  be 
subordinate  ;  and  a  species  in  relation  to  a  species  is  said  to 
be  co-ordinate.  Thus,  in  the  division  of  man  into  Caucasians, 
Mongolians,  etc.,  "  man  "  is  superordinate  in  prior  relation  to 
Caucasians,  Mongolians,  etc.,  and  they  are  subordinate  to  man 
in  ulterior  relation,  while  Caucasians,  Mongolians,  etc.,  in  re- 
lation to  each  other,  are  co-ordinate. 

We  may  adopt,  as  is  apparent  in  this  outline,  a  new  princi- 
ple for  each  successive  process  of  division.  In  the  division  of 
plane  and  solid  figures,  however,  it  is  the  same.  In  the  first 
division  it  is  the  form  in  general  ;  in  the  second  division  it  is 
the  kind  of  bounding  lines  ;  in  the  third  division  it  is  the  dif- 


Solid   A 


DEFINITION  AND  DIVISION 


li- 


ferent relative  positions  and  relations  of  lines.  An  illustration 
of  a  completely  new  principle  in  each  division  will  be  the  fol- 
lowing. I  do  not  pretend  that  it  is  perfect,  but  only  that  it 
illustrates  the  point  under  consideration  : 


Mechanical 
Physical  -J 


Science  -J 


Moral 


j  Physics. 
I  Chemistry, 
j  Biology. 
'{  Physiology. 
J  History. 
j  Sociology. 
I  Noetics. 
Psychological  -  .-Esthetics. 
I  Ethics. 


j   Organic 
[  Political 


The  simplest  form  of  logical  division  is  called  Dichotomy, 
which  is  the  continual  division  of  a  genus  into  two  species,  a 
positive  and  a  negative.  This  is  the  simplest  mode  of  making 
the  division  exhaustive.  A  threefold  division  is  called  Tri- 
chotomy ;  but  there  is  no  technical  name  for  the  forms  after 
that.  Dichotomy  is  very  useful  in  certain  kinds  of  discussion, 
but  in  other  circumstances  is  not  so  convenient.  An  example 
of  it  is  found  in  what  is  called  the  Tree  of  Porphyry,  named 
after  the  Greek  logician  who  originated  it.  It  may  be  repre- 
sented thus : 

Substance. 


Corporeal.  Incorporeal. 

Body. 
Animate.  Inanimate. 

Living  being. 
Sensible.  Insensible. 


Animal. 
Rational.  Irrational. 

Man. 
Socrates.        Plato.      Aristotle  and  others. 


98  ELEMENTS  OF  LOGIC 

Man  could  be  dichotomously  divided  into  Caucasian  and 
non-Caucasian,  the  former  into  Greeks  and  not-Greeks,  and 
so  on.  But  the  process  must  be  terminated  in  the  last  analysis 
with  a  mention  of  the  individuals,  and  there  may  be  several 
points  where  this  may  be  legitimately  done. 

The  usefulness  of  dichotomy  has  its  limitations.  Thus  it 
would  be  useless  to  divide  Europe  into  France  and  not-France, 
the  British  Empire  into  England  and  not-England,  or  America 
into  Rhode  Island  and  not-Rkode  Island.  "  Dichotomy  is  use- 
less and  even  seems  absurd  in  these  cases,  because  we  can  ob- 
serve the  rules  of  division  certainly  in  a  much  briefer  division. 
But  in  less  certain  branches  of  knowledge  our  divisions  can 
never  be  free  from  possible  oversight  unless  they  proceed  by 
dichotomy.  Thus,  if  we  divide  the  population  of  the  world 
into  Aryan,  Semitic,  and  Turanian,  some  race  might  ultimately 
be  discovered  which  is  distinct  from  any  of  these,  and  for 
which  no  place  has  been  provided ;  but  had  we  proceeded 
thus: 

Man 

I 


I  I 

Aryan  Not- Aryan 


Semitic  Not-Semitic 

I 


I  I 

Turanian  Not-Turanian, 

it  is  evident  that  the  new  race  would  fall  into  the  last  group, 
which  is  neither  Aryan,  Semitic,  nor  Turanian.  All  the  divis- 
ions of  naturalists  are  liable  to  this  inconvenience.  If  we  di- 
vide Vertebrate  Animals  into  Mammalia,  Birds,  Reptiles,  and 
Fish,  it  may  happen  at  any  time  that  a  new  form  is  discovered 
which  belongs  to  none  of  these,  and  therefore  upsets  the  di- 
vision." 

Jevons  might  have  remarked  that  dichotomy  can  be  best  ap- 
plied where  our  knowledge  has  not  been  exhausted,  or  where 
changes  of  boundary  are  likely  to  take  place.  Where  there  are 
distinct  and  known  limits  to  a  body  of  knowledge,  the  genus 


DEFINITION  AND  DIVISION  99 

can  be  more  clearly  exhausted  by  the  usual  method  of  division. 
Dichotomy  is  needed  where  there  is  an  indistinct  and  uncertain 
field  of  ideas. 

2.  Partition. — Partition  is  the  analysis  of  a  concept  by  a 
statement  of  its  intension.  It  is  simply  a  process  describing 
the  concept  by  its  qualities,  or  parts  constituting  it.  The  con- 
cept may  be  viewed  either  as  an  individual  or  a  class-whole, 
and  partition  merely  defines  it  by  its  properties.  Partition 
may  be  mathematical  or  quantitative  and  logical  or  qualitative. 
It  is  mathematical  when  the  division  or  analysis  is  into  its 
parts  expressed  in  terms  of  space  or  time.  Thus  the  concept 
tree  is  partitioned  mathematically  into  roots,  trunk,  branches, 
and  leaves  ;  logically,  it  would  be  divided  into  its  vegetable 
properties — color,  woody  fibre,  raising  of  the  sap  by  capillary 
attraction,  etc.  The  concept  "  life  "  (a  person's  age)  would  be 
partitioned  mathematically  into  childhood,  maturity,  old  age, 
etc.  ;  logically,  it  might  be  divided  or  partitioned  into  its 
length,  goodness  or  badness,  mode  of  spending  it,  etc.  But 
in  all  cases  we  should  disregard  the  questions  of  genus  and 
species,  and  merely  endeavor  to  consider  the  properties,  essen- 
tial or  accidental,  which  constitute  a  concept.  To  make  the 
partition  exhaustive  we  should  be  obliged  to  state  all  the 
properties  that  make  up  an  object ;  but  as  this  is  often  pre- 
vented by  the  limitations  of  our  knowledge  of  them,  we  have 
to  be  content  with  such  as  we  know.  A  more  complete  illus- 
tration of  the  analysis  than  those  given  may  better  show  the 
extent  to  which  it  may  be  carried.  Take  the  case  of  "  gold." 
The  qualities  of  it  are  that  it  is  material,  metallic,  solid,  ele- 
mentary, yellow,  malleable,  precious,  useful,  conductor  of  elec- 
tricity, etc.  "  Man"  again  may  be  partitioned  into  animality, 
rationality,  color,  weight,  sociality,  etc. 

The  usefulness  of  the  process  is  not  so  apparent  in  common 
conceptions.  But  if  it  were  carried  out  carefully  with  such 
conceptions  as  "virtue,"  "thought,"  "mind,"  "religion," 
"  cause,"  "  intuitive,"  "  law,"  "nature,"  etc.,  many  a  controversy 
would  be  modified  in  its  incidents.  Thus  the  notion  "  cause  " 
may  be  partitioned  into  uniformity  of  sequence  and  co-exist- 


100  ELEMENTS  OF  LOGIC 

tence,  and  efficiency  of  power,  and  perhaps  other  qualities. 
Controversies  about  our  knowledge  of  it  would  be  materially 
affected  by  the  presence  or  the  absence  of  the  second  quality. 
Usually  the  partition  implies  that  an  object  has  more  than 
one,  or  that  a  concept  represents  more  than  one  quality.  But 
we  may  regard  every  simple  descriptive  or  declarative  jDroposi- 
tion  as  a  case  of  partition.  It  is  a  descriptive  definition,  which 
partition  aims  to  give.  But  it  is  not  necessary  to  carry  the 
idea  of  partition  so  far,  except  to  intimate  the  broad  distinc- 
tion between  it  and  true  logical  definition.  Partition,  where 
it  serves  any  useful  purpose,  assumes  a  multiple  of  qualities 
which  require  recognition  as  well  as  the  relation  of  genus  and 
species.  It  is  opposed  to  division  as  intension  is  opposed  to 
the  extension  of  concepts.  Hence  it  is  a  process  complement- 
ary to  division. 

3d.  Definition  of  Concepts. — The  definition  of  anything 
in  practice  is  undertaken  in  several  ways.  But  Logic  has 
strictly  to  do  with  only  one  of  them.  This  is  called  Logical 
Definition.  It  is  necessary,  however,  to  notice  the  several 
modes  of  definition  in  order  to  distinguish  the  logical  from 
them  in  the  proper  way.  They  may  be  called  Etymological 
Definition,  Descriptive  Definition,  and  Logical  Definition. 

1.  Etymological  Definition. — This  is  the  definition  of  a 
concept  by  the  word-roots  from  which  the  term  originated. 
For  example,  "  inquisition  "  would  be  etymologically  defined 
by  saying  it  was  from  two  roots  or  words  denoting  "  to  in- 
quire into  ; "  "  playfulness,"  from  a  root  and  suffixes  denoting 
the  quality  of  being  full  of  sport,  etc.  But  while  this  form  of 
definition  is  very  valuable  in  some  circumstances  it  is  of  no 
importance  in  logical  doctrine,  or  for  elucidating  any  of  the 
laws  of  thought. 

2.  Descriptive  Definition. — This  is  the  definition  of  a  con- 
cept or  a  thing  merely  by  describing  it,  and  is  essentially  the 
same  as  partition.  It  most  frequently  occurs  in  the  mention 
of  accidental  properties  of  objects  when  distinguished  from 
complete  partition  and  definition  proper.  It  is,  in  fact,  im- 
perfect definition  in  the  omission  of  one  or  the  other  of  the 


DEFINITION  AND  DIVISION  101 

two  essential  conditions  of  the  logical  form  of  it.  Thus,  a 
descriptive  definition  of  a  triangle  would  be,  that  it  is  com- 
posed of  straight  hues  and  a  symbol  very  frequently  used  in 
geometry.  But  such  an  account  of  it  might  as  well  apply  to 
a  rectangle  or  a  parallelogram.  A  descriptive  definition,  how- 
ever, may  be  made  to  approximate  very  closely  to  the  logical. 
The  above  illustration  wants  but  little  modification  to  become 
that.  Indeed  the  true  definition  is  descriptive,  but  it  is  com- 
pletely descriptive,  involving  a  relation  not  expressed  in  this 
imperfect  form.  The  difference  between  them  can  perhaps 
be  technically  expressed  by  saying  that  the  ordinary  descrip- 
tive definition  depends  upon  the  accidentia  or  the  conferentia, 
and  logical  definition  upon  the  differentia. 

3.  Logical  Definition. — Logical  definition  is  the  statement 
of  the  genus  and  differentia  of  a  concept,  and  is  thus  occupied 
with  the  whole  of  its  intension,  as  Division  is  occupied  with 
the  whole  of  its  extension.  To  illustrate,  I  may  define  "  man  " 
as  a  "rational  animal."  In  this  statement  "animal"  is  the 
genus  to  which  "  man  "  is  supposed  to  belong  as  a  species, 
and  rationality  is  the  differentia  which  distinguishes  him  from 
other  species.  Again,  I  may  define  a  circle  as  "  a  curved  line 
everywhere  equally  distant  from  a  point  within  called  the  cen- 
tre ;"  or  tree  as  "a  vegetable  with  woody  fibre,  root,  trunk, 
branches,  leaves,  and  a  certain  magnitude  ;"  or  a  house  as  "a 
building  used  for  a  place  of  residence,"  etc.,  and  in  each  case 
fulfil  the  requirements  of  a  logical  definition. 

It  is  important  to  make  two  remarks  in  regard  to  these  and 
all  logical  definitions.  First,  they  do  not  state  the  mathemat- 
ical genus  and  species,  which  would  identify  the  process  with 
division  in  its  principles,  but  the  conferentia  or  logical  genus, 
and  the  differentia.  Second,  it  is  always  the  species  that  is  de- 
fined, never  the  genus. 

The  difference  between  definition  and  division,  in  their 
treatment  of  the  genus,  is,  that  the  division  of  a  concept  pro- 
ceeds progressively  to  the  species  under  it,  and  may  continue 
on  down  to  the  individual,  but  definition  proceeds  regressively 
to  the  proximate  genus,  or  some  appropriate  genus  which  may 


102  ELEMENTS  OF  LOGIC 

serve  as  such,  and  goes  no  farther  unless  called  upon  to  define 
that  in  the  same  way.  Its  progressive  movement,  if  it  can  be 
said  to  have  any,  is  a  statement  of  the  differential  property  or 
properties  of  the  species  defined,  not  a  development  of  its  ex- 
tension or  subordinate  species.  Since  it  is  the  species,  there- 
fore, which  is  always  defined,  and  never  the  genus,  unless  it 
be  also  a  species  in  relation  to  a  higher  genus,  the  ultimate  or 
summum  genus  can  never  be  defined.  A  logical  definition,  as  we 
have  shown,  must  state  the  genus  ;  but  as  the  summun  genus  is 
never  a  species  it  cannot  be  logically  defined.  Hence  the  impos- 
sibility of  defining  ultimate  truths  or  principles,  or  the  sinqilest 
concepts.  They  can  be  dealt  with  only  descriptively  or  par- 
titively,  or  divisively.  Everything  below  them  can  be  defined. 
But  it  is  important  to  observe  the  meaning  of  stating  the 
"genus  and  differentia"  in  a  logical  definition.  We  observe, 
first,  that  it  is  not  the  mathematical  "  genus  and  species,"  as 
already  remarked.  Hence  the  term  genus  is  used  in  its  mean- 
ing as  contrasted  with  differentia,  and  so  denotes  the  logical 
as  opposed  to  the  mathematical  genus.  It  therefore  denotes 
the  common  qualities,  or  conferentia  of  the  concept  defined,  in 
comparison  with  co-ordinate  species.  This  is  to  show  that 
logical  definition  is  a  statement  of  the  conferentia  and  differ- 
entia of  a  conception,  and  not  of  the  genus  and  species  mathe- 
matically considered.  Hence  we  see  how  it  is  exclusively  oc- 
cupied, if  not  explicitly,  then  inrplicitly,  with  the  intension  of 
concepts.  The  genus,  as  contrasted  with  the  differentia,  de- 
notes only  the  common  qualities  of  objects  in  the  same  class, 
and  the  differentia  those  which  separate  the  individual  or 
species  defined  from  others  in  the  same  class.  Thus  to  de- 
fine a  "bed"  as  "a  piece  of  furniture  for  reclining  upon,"  is 
to  intimate  that  "bed"  has  qualities  in  common  with  other 
things  known  as  furniture,  and  it  is  distinguished  from  a 
chair,  which  is  used  for  sitting,  by  the  property  of  being  used 
for  reclining.  Instead  of  specifying  the  common  qualities  par- 
titively,  however,  a  general  name  suffices  to  imply  them.  But 
not  having  any  corresponding  abstract  term  for  differentia, 
these  properties  have  to  be  distinctly  indicated.     The  genus 


DEFINITION  AND  DIVISION  103 

and  differentia,  therefore,  as  stated  in  the  definition,  are  sim- 
ply all  the  qualities  that  make  up  the  species  defined.  These 
are  the  conferentia  and  the  differentia. 

The  conclusion  from  this  must  be  that  the  predicate  of  a 
logical  definition  is  always  equal  to,  identical,  and  convertible 
with  the  subject.  The  importance  of  this  will  be  apparent  in 
the  doctrine  of  Conversion  and  of  Reasoning.  All  that  we  re- 
quire to  observe  here  is  the  difference  between  the  ordinary 
simple  proposition  and  the  proposition  which  is  regarded  as  a 
definition.  The  statements  that  "man  is  a  biped,"  or  "  man  is 
mortal,"  are  not  definitions.  One  states  merely  the  genus  of 
man,  and  the  other  a  property  of  him.  The  first  implies  a 
property,  that  of  two-footedness,  and  the  second  may  be  said 
to  imply  a  genus.  But  neither  of  them  specifies  any  differen- 
tia that  would  distinguish  man,  on  the  one  hand,  from  other 
bipeds,  and,  on  the  other  hand,  from  other  mortals.  They  are 
not  definitions  because  they  do  not  state  the  whole  intension 
of  the  species.  Hence  a  true  definition  expresses  the  full 
meaning  of  the  species  defined,  and  can  be  used  convertibly 
with  it.  Thus  we  can  say  equally  that  "  man  is  a  rational 
animal,"  or  "  rational  animals  are  men,"  a  process  which  can- 
not be  performed  with  the  simple  proposition  unless  it  is  con- 
sidered a  definition.  The  fact  that  the  two  forms  of  propo- 
sition are  the  same  to  all  api)earances,  and  the  fact  that  the 
mind  uses  the  subject  and  predicate  of  definitions  convertibly 
with  each  other,  often  lead  to  confusion,  by  inducing  the  treat- 
ment of  simple  propositions  as  definitions.  This  source  of  er- 
ror will  be  treated  in  its  proper  place.  At  present  it  suffices 
to  call  attention  to  the  fact. 

In  regard  to  the  rules  regulating  correct  definitions  it  will 
suffice  to  state  Jevons's  account  of  them,  and  it  will  always  be 
important  for  the  student  to  keep  them  in  mind.    They  are  five  : 

(a)  "  A  definition  should  state  the  essential  attribute*  of  the 
species  defined.  So  far  as  any  exact  meaning  can  be  given  to 
the  expression  '  essential  attributes,'  it  means  the  proximate 
genus  and  difference." 

(b)  "  A  definition  must  not  contain  the  name  defined.     For 


104  ELEMENTS  OF  LOGIC 

the  purpose  of  the  definition  is  to  make  the  species  known, 
and  as  long  as  it  is  not  known  it  cannot  serve  to  make  itself 
known.  When  this  rule  is  not  observed,  there  is  said  to 
be  a  '  circulus  in  definiendo,'  or  '  circle  in  definition,'  because 
the  definition  brings  us  around  again  to  the  very  word  from 
wThich  we  started.  This  fault  will  usually  be  committed  by 
using  a  word  in  the  definition  which  is  really  a  synonym  of 
the  name  defined,  as  if  I  were  to  define  a  '  plant  'as  'an  or- 
ganized being,  possessing  vegetable  life,'  or  '  elements '  as 
'  simple  substances,'  vegetable  being  really  equivalent  to  plant, 
and  simple  to  elementary.  If  I  were  to  define  '  metals '  as 
'  substances  possessing  metallic  lustre,'  I  should  either  commit 
this  fault  or  use  the  term  metallic  lustre  in  a  sense  which 
would  admit  other  substances  and  thus  break  the  folio  wing  rule." 
(e)  "  The  definition  must  be  exactly  equivalent  to  the  species 
defined.  That  is  to  say,  it  must  be  an  expression,  the  denota- 
tion of  which  is  neither  narrower  nor  wider  than  the  species,  so 
as  to  include  exactly  the  same  objects.  The  definition,  in  short, 
must  denote  the  species,  and  nothing  but  the  species,  and  this 
may  really  be  considered  a  description  of  what  a  definition  is." 

(d)  "A  definition  must  not  be  expressed  in  obscure,  figurative, 
or  ambiguous  language.  In  other  words,  the  terms  employed  in 
the  definition  must  be  all  exactly  known,  otherwise  the  pur- 
pose of  the  definition,  to  make  us  accpaainted  with  the  sufficient 
marks  of  the  species,  is  obviously  defeated.  There  is  no  worse 
logical  fault  than  to  define  ignotum  per  ignotius,  the  unknown 
by  the  still  more  unknown.  Aristotle's  definition  of  the  soul 
as  '  the  entelechy,  or  first  form  of  an  organized  body  which 
has  potential  life,'  certainly  seems  subject  to  this  objection." 

(e)  "  A  definition  must  not  be  negative  when  it  can  be  affirm- 
ative. This  ride,  however,  is  often  not  applicable,  and  is  by  no 
means  always  binding."  * 

*  The  following  references  may  be  consulted  on  matters  pertaining  to 
this  chapter  :  Mill :  Logic,  Book  I.,  Chaps.  VII.  and  VIII.  ;  Venn  :  Em- 
pirical Logic,  Chaps.  XI.  and  XII.;  Hamilton:  Lectures  on  Logic,  Lects. 
XXIV.  and  XXV.  ;  De  Morgan:  Formal  Logic,  Chap.  XII.  ;  Whately: 
Elements  of  Logic,  Book  II.,  Chap.  V.;    .Supplement  to  Chap.  I.,  §§  2-G. 


CHAPTER  VTL 
PROPOSITIONS  OR  JUDGMENTS 

1st.  Definition. — Words  or  terms  unconnected  express 
only  concepts  outside  of  any  distinctly  affirmed  relation.  In 
this  way  they  do  not  convey  truth,  but  only  ideas  or  concep- 
tions. Logic  has  to  deal  with  the  connection  of  concepts  and 
their  implications.  The  manner  in  which  terms  and  concepts 
are  joined  together  determines  what  a  proposition  shall  be. 
It  is  not  every  combination  of  terms  that  forms  a  logical  pi-op- 
osition.  Some  combinations  may  be  mere  phrases  or  ejacula- 
tions. But  those  combinations  expressing  a  certain  kind  of 
relation,  namely,  a  declarative  relation  between  two  terms,  are 
the  propositions  with  which  Logic  is  concerned. 

A  proposition  in  Grammar  is  called  a  sentence  ;  in  Logic,  it 
is  most  frequently  called  a,  judgment.  A  proposition  or  judg- 
ment, therefore,  in  Logic,  is  the  affirmation  or  denial  of  agree- 
ment between  two  conceptions.  It  involves  a  comparison  be- 
tween them  and  a  perception  of  this  relation.  Thus  the 
proposition,  "  Gold  is  a  metal,"  expresses  a  certain  agreement 
between  the  concepts  "gold"  and  "metal,"  an  agreement 
which  implies  that  the  same  quality  is  common  to  both,  or 
that  "gold"  is  a  species  of  "metal."  On  the  other  hand,  the 
proposition,  "Man  is  not  a  quadruped,"  expresses  a  disagree- 
ment in  a  certain  particular  between  the  two  concepts — a  dis- 
agreement which  implies  that  "  man "  is  not  in  the  class 
"  quadrupeds,"  or  does  not  possess  the  particular  quality 
which  distinguishes  quadrupeds.  This  agreement  or  disagree- 
ment is  not  limited  to  single  concepts  or  temis,  but  may  in- 
clude the  same  relation  between  groups  of  concepts  constitut- 
ing phrases.  Thus,  "  The  City  of  Washington,  in  the  District 
of  Columbia,  is  the  Capital  of  the  United  States  of  America," 


106  ELEMENTS  OF  LOGIC 

is  a  proposition,  only  a  little  more  complex  in  its  elements 
than  the  former  illustrations. 

The  terms  between  which  the  relation  is  asserted  or  denied 
are  called  the  subject  and  the  predicate.  The  subject  is  that 
of  which  something  is  affirmed  or  denied  ;  the  predicate  is 
that  which  is  affirmed  or  denied  of  the  subject.  "Subject" 
(subjectum,  v-rroKelfjievov)  means  underlying  thing  ;  "  predicate  " 
(prsedicatum,  KaTwyopovfxevov)  means  that  which  is  asserted. 
The  subject  and  predicate  may  be  either  grammatical  or  logi- 
cal. The  grammatical  subject  or  predicate  will  be  a  single 
term  ;  the  logical  subject  and  predicate  will  consist  of  the 
grammatical  subject  with  all  its  modifiers.  Taken  together 
they  express  in  thought  a  single  idea  or  conception,  and  hence 
Logic  may  treat  them  accordingly.  All  complex  propositions 
are  thus  reduced  to  a  single  form. 

The  term  expressing  the  connection  between  the  subject  and 
predicate  is  called  the  copula,  and  is  always  some  form  of  the 
verb  to  be,  or  its  equivalent.  In  many,  perhaps  the  largest 
number  of  propositions,  the  verb  to  be  is  not  found,  and  hence 
they  appear  to  be  wanting  in  a  copula.  Thus  the  proposition, 
"  Napoleon  ruled  France,"  contains  no  expressed  copula.  In 
all  such  propositions,  however,  the  predicate  is  said  to  include 
the  verb  and  its  dependent  terms,  and  so  to  include  the 
suppressed  copula.  Thus  in  the  illustration  given,  "  ruled 
France  "  is  called  the  predicate,  and  the  proposition  seems  to 
consist  of  only  subject  and  predicate.  But  if  we  resolve  the 
expression  "  ruled  France  "  into  its  exact  logical  equivalent, 
"  was  the  ruler  of  France,"  we  have  the  copula  and  the  predi- 
cate in  the  simple  form.  Hence  the  term  "  France  "  will  not 
be  the  predicate  alone,  but  "  ruled  France  "  must  represent  it 
with  the  copula  implied  or  included  in  it.  It  is  necessary  to 
so  consider  the  matter  in  order  to  deal  logically  with  all  such 
propositions.  This  logical  treatment  of  them  depends  upon 
such  a  conception  of  the  relation  between  subject  and  predi- 
cate as  can  be  reduced  to  a  general  or  universal  law.  A  more 
complete  discussion  of  the  nature  of  this  relation  will  be  ap- 
propriate after  we  have  considered  the  divisions  of  judgments. 


PROPOSITIONS  OR  JUDGMENTS  107 

2d.  Divisions. — Propositions  can  be  divided  in  a  great  many 
ways.  The  first  division  into  Indicative,  Interrogative,  and  Im- 
perative, with  perhaps  the  Optative  and  Exclamatory,  as  recog- 
nized by  some,  is  grammatical,  and  it  is  only  with  the  first  class 
that  Logic  has  to  do.  The  essential  meaning  of  the  others,  s<  > 
far  as  the  relation  of  concepts  is  concerned,  can  be  reduced  to 
the  first  form,  the  declarative,  or  indicative  in  such  emergen- 
cies as  require  a  logical  use  of  their  matter. 

1.  Logico-Gkammatical  Propositions. — There  is  a  second  di- 
vision which  is  both  grammatical  and  logical,  but  which  has 
not  been  uniformly  the  same  with  logicians.  Sometimes  it  has 
been  into  Categorical  and  Conditional,  with  a  subdivision  of 
the  second  into  Hypothetical  and  Disjunctive.  Sometimes  Con- 
ditional and  Hypothetical  simply  interchange  places  in  this 
division,  and  in  a  third  form  they  are  made  synonymous  with 
each  other,  giving  us  a  co-ordinate  division  of  three  kinds, 
into  Categorical,  Conditional  or  Hypothetical,  and  Disjunctive. 
This  last  division  I  much  prefer  to  all  others,  for  the  reason 
that  their  relation  to  each  other  in  structure  and  meaning  can 
be  more  easily  determined  than  in  any  other  classification. 

There  are,  however,  two  classifications  which  may  be  given 
and  that  are  of  considerable  convenience  in  explaining  the 
meaning  and  relations  of  various  kinds  of  propositions.  The 
first  one  proceeds  in  the  order  of  increasing  complexity,  and 
is  intended  to  mark  the  nature  of  the  additions  made  to  deter- 
mine the  more  complex  forms.  The  following  diagram  ex- 
hibits the  classification  with  illustrations  : 

{n  .         •     i  (  Declarative  =  A  is  B. 

categorical  -  r)jsjunctivc  =  a  is  either  B  or  C. 

n      ,...       .  Hypothetical  =  If  A  is  B,  C  is  D. 

Conditional  j  Dfiemmatio  =If  A  ia  H,  C  is  either  D  or  E. 

It  is  apparent  in  this  division  that  the  simplest  form  is  the 
declarative  proposition,  where  the  assertion  is  absolute  and 
definite  as  regards  both  subject  and  predicate.  The  disjunc- 
tive form  is  equally  assertory  in  its  form  of  expression,  but 
differs  from  the  first  in  allowing  some  doubt  or  choice  about 
the  predicate,  there  being  one  alternative  which  excludes  the 


108  ELEMENTS  OF  LOGIC 

connection  of  the  other  with  the  subject.  The  hypothetical 
expresses  a  definite  dependence  of  one  proposition,  a  declara- 
tive proposition  upon  a  condition.  Hence  it  adds  a  declara- 
tive assertion  to  a  conditional  one.  The  dilenimatic  imposi- 
tion simply  adds  a  disjunctive  one  to  a  conditional  proposi- 
tion. 

But  a  second  classification  is  much  preferable  to  this  because 
it  conforms  to  the  three  forms  of  reasoning,  and,  in  a  measure, 
determines  them.  In  this  classification  we  make  the  disjunc- 
tive appear  as  co-ordinate  with  the  other  two,  although  it  is  in 
reality  a  combination  of  them,  and  to  which  we  apply  the  dis- 
tinction between  form  and  matter.  The  categorical  and  con- 
ditional propositions  are  regarded  logically  as  pure  and  un- 
mixed. The  disjunctive  we  make  categorical  in  its  form  of 
expression,  but  conditional  in  its  meaning.  This  the  following 
diagram  will  show : 

l  Categorical  =  Assertory  in  form  and  matter. 
Propositions  <  Conditional  =  Hypothetical  in  form  and  matter. 

(  Disjunctive  =  Categorical  in  form,  but  Conditional  in  matter. 

In  regard  to  the  disjunctive  proposition,  under  this  conception 
of  it,  it  need  only  be  said  that  its  form  of  expression  is  un- 
doubtedly assertory.  It  is  j>ositively  affirmed  that  "A  is  either 
B  or  C."  But  the  meaning  of  the  disjunction,  or  the  alterna- 
tive expressed,  can  be  understood  only  as  implying  "  if  A  is  B, 
it  is  not  C,"  or  "  if  A  is  C,  it  is  not  B."  We  shall  discover 
later  on  in  the  discussion  of  reasoning  that  this  is  the  only  in- 
terpretation of  the  case  which  will  enable  us  to  reduce  disjunc- 
tive reasoning  to  the  regular  form,  or  to  understand  it  as  a 
mode  of  the  usual  process  of  reasoning. 

A  Categorical  proposition  is  one  in  which  a  statement  is  un- 
conditionally made  ;  as,  "  A  is  B,"  or,  "  Man  is  mortal."  A  Con- 
ditional or  Hypothetical  proposition  is  one  in  which  the  asser- 
tion is  conditional  or  dependent  upon  a  supposition  of  some 
kind  ;  as,  "If  A  is  B,  C  is  D,"  or,  "If  a  stone  be  released  from 
support  it  will  fall  to  the  ground."  The  first  clause  of  the 
conditional  proposition  is  called  the  antecedent,  the  second  the 
consequent.     The  symbols  of  such  propositions  are  if,  even  if, 


PROPOSITIONS  OR  JUDGMENTS  10'J 

provided  that,  although,  sometimes  when,  or  any  form  of  ex- 
pression denoting  a  condition.  A  Disjunctive  proposition  is 
one  which  implies  or  asserts  an  alternative  in  the  relation  be- 
tween the  subject  and  predicate  ;  as,  "  A  is  either  B  or  C,"  or, 
"Metals  are  either  hard  or  soft."  The  symbols  of  the  disjunc- 
tive proposition  are  either  and  or.  Some  ambiguity  is  con- 
nected with  their  meaning,  which  will  have  to  be  considered 
when  discussing  the  Disjunctive  Syllogism.  But  as  it  does 
not  effect  the  form  and  general  meaning  of  the  proposition  by 
that  name,  the  matter  need  not  be  discussed  at  present.  We 
have  only  to  remark  what  the  disjunction  means  when  it  is 
complete,  and  that  is,  that  the  alternatives  expressed  by  the 
terms  either  and  or  should  be  exhaustive.  It  means  that  the 
connection  between  the  subject  and  predicate  must  be  one  or 
the  other  of  two  things.  In  the  proposition  A  is  either  B  or  C, 
the  question  whether  A  is  B  or  A  is  C  is  indefinite  or  unde- 
cided, but  it  is  definitely  one  or  the  other,  and  hence  the 
proposition  either  means  that  A  is  B  and  not  C,  or  it  means 
that  A  is  G  and  not  B.  Hence,  although  the  proposition  stands 
as  a  direct  assertion,  it  means  that  if  A  is  B,  it  is  not  C,  or  if  A 
is  C,  it  is  not  B.  This  is  the  reason  that  it  is  usually  classed  as 
a  form  of  conditional  judgment.  But  if  it  be  closely  examined 
it  will  be  found  to  contain  both  assertory  and  conditional  ele- 
ments. It  is  categorical  in  its  form,  and  conditional  in  its 
matter  or  meaning. 

2.  Propositions  According  to  Quality. — Propositions  may 
be  divided  into  Affirmative  and  Negative,  according  as  they 
affirm  or  deny  the  agreement  between  the  subject  and  the 
predicate.  This  relation  is  called  or  determines  their  quality. 
An  affirmative  proposition  asserts  an  agreement  between  sub- 
ject and  predicate  ;  as,  "  Gold  is  yellow,"  or,  "  Doves  are  birds." 
A  negative  proposition  is  one  which  denies  an  agreement  be- 
tween subject  and  predicate  ;  as,  "  Men  are  not  trees,"  or, 
"Gas  is  not  heavy." 

3.  Propositions  According  to  Quantity. — Propositions  ac- 
cording to  quantity  are  divided  into  Universal  and  Particular. 
The  distinction  between  them  is  determined  by  the  question 


110  ELEMENTS  OF  LOGIC 

whether  the  predicate  is  affirmed  or  denied  of  the  whole  of  the 
subject.  Hence  a  universal  proposition  is  one  in  which  the 
predicate  is  said  to  be  affirmed  or  denied  of  the  whole  of  the 
subject ;  as,  "  All  men  are  mortal,"  or,  "  No  men  are  trees."  A 
particular  proposition  is  one  in  which  the  predicate  is  said  to 
be  affirmed  or  denied  of  a  part  of  the  subject ;  as,  "  Some 
men  are  wise,"  or,  "  Some  snow  is  not  black."  But  the  diffi- 
culty with  this  definition  is  that  there  is  a  sense  in  which 
the  predicate  is  affirmed  or  denied  of  the  whole  of  the  sub- 
ject in  the  particular  proposition.  For  according  to  what 
has  been  said  of  the  nature  of  the  subject  it  may  include  what 
is  known  in  grammar  as  the  "  logical  subject,"  which  consists  of 
all  the  terms  constituting  a  complex  conception  and  standing 
in  the  relation  of  "  subject  "  to  the  proposition.  In  this  sense 
the  predicate  of  a  particular  proposition  is  affirmed  or  denied 
of  the  whole  of  its  logical  subject,  but  of  only  a  part  of  the 
grammatical  subject.  If  therefore  we  could  say  that  a  uni- 
versal proposition  affirms  or  denies  the  predicate  of  the  whole 
subject,  grammatical  and  logical,  and  a  particular  proposition, 
of  a  part  of  the  grammatical  subject  only,  the  difficulty  would 
be  removed.  But  it  returns  again  in  such  propositions  as 
"  All  good  men  are  respected,"  which  would  be  particular  ac- 
cording to  the  definition.  For  the  predicate  is  affirmed  of 
only  a  part  of  the  grammatical  subject. 

It  would,  therefore,  be  better  for  the  purposes  of  definition 
either  to  divide  propositions  into  Definite  and  Indefinite,  or 
define  universal  propositions  as  affirming  or  denying  the  pred- 
icate of  the  whole  of  a  definite  subject,  and  particular  proposi- 
tions, of  an  indefinite  subject.  This  is  what  is  really  meant  by 
universal  and  particular  propositions,  and  hence,  with  the  pro- 
viso that  they  shall  be  identical  in  meaning  with  definite  and 
indefinite,  we  shall  adopt  them  as  expressing  the  division  of 
propositions  according  to  quantity. 

But  this  twofold  division  is  the  result  of  a  reduction  from 
a  division  which  is  frequently  fivefold.  Propositions  are  fre- 
quently divided,  according  to  quantity,  into  Universal,  Singu- 
lar, General,  Pluralive,  and  Particular.     The  first  and  the  last 


PROPOSITIONS  OR  JUDGMENTS  111 

have  been  adequately  defined  and  illustrated.  The  intermediate 
three  may  be  reduced  to  one  or  the  other  of  the  first  and  the 
last,  as  their  definition  will  prove.  A  singular  proposition  is 
one  in  which  the  subject  is  a  singular  term,  and  hence  definite 
in  its  meaning  ;  as,  "  Louis  XIV.  was  king  of  France."  Here 
the  predicate  is  affirmed  of  the  whole  of  a  definite  subject,  and 
hence  for  all  logical  purposes  the  proposition  is  universal. 
That  is,  the  same  laws  of  reasoning,  mediate  or  immediate, 
will  apply  to  singular  or  apply  to  universal  propositions.  A 
general  proposition  is  one  in  which  the  extension  of  the  sub- 
ject is  ambiguous;  as,  "Metals  are  useful,"  "Man  is  intelli- 
gent." It  is  not  stated  whether  "  All  metals  are  useful,"  or 
"  All  men  are  intelligent,"  or  whether  some  are  so.  The  prop- 
ositions are  capable  of  either  interpretation,  and  according  as 
we  think  of  all  or  some,  are  universal  or  particular.  Plurative 
propositions  are  undoubtedly  particular.  They  are  intro- 
duced by  the  word  most,  or  its  equivalent ;  as,  "  Most  rumi- 
nants are  horned,"  and  require  mention  only  because  of  a  pe- 
culiar syllogism  which  is  valid  in  spite  of  its  composition  from 
particular  premises  ;  of  which  again.  But  they  affirm  or 
deny  the  predicate  definitely  of  more  than  half  the  subject,  but 
indefinitely  in  regard  to  which  of  the  two  halves  is  exhausted 
in  the  term  most.  They  are,  therefore,  classed  as  particular 
propositions.  A  summaiy  of  this  reduction  appears  in  the 
following  table  : 

(  t  Universal    ) 

j  Definite    <  Singular      J-  Universal. 

Propositions  -!  ,  General  ( 


|  Indefinite  -f  Plurative    >  Particular. 
[  (  Particular  ) 

The  mark  of  a  universal  proposition  usually  consists  of  some 
adjective  denoting  quantity,  such  as,  all,  every,  each,  any 
(meaning  all  individually),  and  whole.  But  wherever  we  find 
the  predicate  referring  definitely  to  the  whole  of  the  subject 
we  may  treat  the  proposition  as  universal.  This  merely  im- 
plies that  some  propositions  may  be  universal  in  their  matter, 
but  indefinite  in  their  form.     The  signs  of  particular  propo- 


112  ELEMENTS  OF  LOGIC 

sitions  are  also  certain  adjectives  of  quantity,  such  as  some, 
certain,  a  few,  many,  most,  any  (meaning  an  indefinite  indi- 
vidual), or  such  others  as  denote  at  least  a  part  of  a  class. 

The  signs  of  a  negative  proposition  are  no,  not,  and  none, 
the  first  and  last  being  prefixed  to  the  subject,  and  the  second 
joined  to  the  copula.  Examples  of  them  are,  "  No  metals  are 
animals,"  "None  of  the  rebels  were  punished,"  and  "  Men  are 
not  quadrupeds."  The  term  "  no  "  is  one  which  denotes  both 
universal  quantity  and  negative  quality  in  propositions. 

The  quality  and  quantity  of  propositions  may  be  combined 
in  classifying  them,  and  we  shall  have  universal  affirmative 
propositions,  universal  negatives,  particular  affirmatives,  and 
particular  negatives.  It  has  been  usual  to  choose  an  abbre- 
viated symbol  to  denote  each  of  these  classes.  The  first  four 
vowels  of  the  alphabet — A,  E,  I,  0 — have  been  chosen  for  this 
purpose.  A  is  the  symbol  of  a  universal  affirmative,  I  of 
a  particular  affirmative,  E  of  a  universal  negative,  and  O  of 
a  particular  negative.  Henceforth  we  shall  employ  them  with 
this  denotation  whenever  it  is  most  convenient.  It  will  be  in- 
teresting to  remark  that  A  and  I  occur  in  the  Latin  affirmo, 
and  E  and  O  in  the  Latin  nego.  There  is  no  significance  in 
this,  save  perhaps  as  a  mnemonic  aid.  The  following  table 
summarizes  results  : 

(Universal    j  A ffirmative  =  A. 
PronnitfnnJ  (Negative       =  E. 

rropoitions  ,  ,  Affirmative  =  I. 

I Partlcular  (Negative      =  O. 

4.  Analytic  and  Synthetic  Peopositions. — Another  division 
separates  propositions  into  Analytic,  Essential  or  Explicative, 
and  Synthetic  or  Ampliative.  An  analytic  proposition  affirms  of 
its  subject  a  predicate,  which  is  implied  in  the  very  conception 
of  the  subject.  Thus,  "  Matter  is  extended,"  "  Water  is  moist," 
"Living  beings  are  organic,"  "Wood  is  a  substance,"  are  all 
analytical  judgments  because  the  subject  cannot  be  represented 
to  the  mind  without  thinking  implicitly  or  explicitly  of  the 
notion  expressed  by  the  predicate.  The  use  of  the  term  "es- 
sential "  to  describe  such  judgments  means  that  the  property 
expressed  by  the  predicate  is  an  essential  one,  which  is  neces- 


PROPOSITIONS  OR  JUDGMENTS  113 

sary  to  conceiving  the  subject.  The  term  "  explicative,"  de- 
scribing the  same  judgment,  means  merely  that  the  predicate 
develops  or  unfolds  what  is  involved  in  the  thought  of  the 
subject.  On  the  other  hand,  a  synthetic  proposition  or  judg- 
ment is  one  in  which  the  predicate  conveys  information  not 
necessarily  implied  in  the  conception  of  the  subject.  Ex- 
amples of  them  are,  "  Water  is  a  conductor  of  sound,"  "  Plato 
was  aristocratic,"  "  Some  men  are  honest,"  "  The  Popes  were 
patrons  of  art."  In  these  instances  the  predicate  is  not  neces- 
sarily associated  with  the  subject.  It  is  no  part  of  our  concep- 
tion of  water  that  it  conducts  sound,  nor  of  the  Popes  that  they 
should  be  patrons  of  art.  It  would  seem  from  this,  therefore, 
that  analytic  judgments  assert  essential  qualities  of  the  sub- 
ject, and  synthetic  judgments  accidental  qualities  of  it.  If  so, 
the  distinction  is  a  very  clear,  and  perhaps  a  very  useful  one. 

But  the  division  of  propositions  into  analytic  and  synthetic 
has  little  or  no  importance  for  Formal  Logic.  Its  chief  im- 
portance is  in  the  domain  of  psychology  and  philosophy.  The 
laws  of  reasoning,  mediate  or  immediate,  are  not  affected  by  it. 
Besides  this  there  is  often  a  great  difficulty  in  distinguishing 
between  the  two  classes  of  j  udgment  so  named,  because  of  the 
confusion  to  which  we  are  liable  in  distinguishing  between  an 
essential  property  which  is  universal,  and  a  universal  property 
which  is  accidental.  Indeed  it  may  be  gravely  doubted 
whether  any  universal  property  can  be  accidental.  At  least 
some  would  doubt  it,  and  it  may  be  a  mere  matter  of  our 
knowledge  as  to  whether  a  given  property  is  essential  or  acci- 
dental. If  so,  the  distinction  between  analytic  and  synthetic 
propositions  will  only  express  the  difference  between  our  mode 
of  representing  a  concept  uniformly  and  the  accidental  asso- 
ciation of  some  other  property  with  it,  less  frequent  in  our  ex- 
perience. That  is,  the  proposition  "  Body  is  extended  "  may 
appear  analytic  to  the  mind  who  has  always  or  most  frequently 
experienced  it  in  connection  with  the  idea  of  extension,  while 
the  want  of  frequent  experience  in  connection  with  its  sono- 
rousness might  make  tbe  proposition  "  Body  is  sonorous  "  a 
synthetic  proposition.     On  the  other  hand,  the  limitation  of 


114  ELEMENTS  OF  LOGIC 

experience  to  hearing  might  make  the  latter  proposition  ana- 
lytic, and  the  former  synthethic. 

It  will  be  seen,  therefore,  that  the  distinction  is  not  only  a 
relative  one,  but  is  mainly  of  psychological  importance.  We 
may,  consequently,  dismiss  it  from  further  consideration. 

5.  Miscellaneous  Propositions. — There  is  a  species  of  prop- 
ositions called  Taulologous  or  Truistic.  They  are  those  which 
affirm  the  subject  of  itself,  and  so  may  be  regarded,  in  form  at 
least,  as  a  kind  of  analytic  judgment.  Thej'  are  such  as  "  A 
is  A,"  "Whatever  is,  is,"  "Man  is  man,"  "A  beast  is  a  beast," 
etc.  Some  of  them,  after  all,  are  synthetic,  and  although  the 
predicate  is  the  same  word  as  the  subject,  it  conveys  a  slightly, 
or  even  wholly,  different  meaning  ;  as,  for  instance,  "  A  man's 
a  man,"  "  The  king  is  king,"  etc.  They  are  tautologous  in  form, 
but  instructive  in  matter.  In  reasoning  we  require  to  be  on 
the  alert  for  such  ambiguity.  Otherwise  the  consideration  of 
truistic  propositions  has  no  logical  importance. 

There  is  another  division  of  propositions  into  Pure  and 
Modal.  "  The  pure  proposition  simply  asserts  that  the  predi- 
cate does  or  does  not  belong  to  the  subject,  while  the  modal 
proposition  states  this  cum  modo,  or  with  an  intimation  of  the 
mode  or  manner  in  which  the  predicate  belongs  to  the  subject. 
The  presence  of  any  adverb  of  time,  place,  manner,  degree, 
etc.,  or  any  expression  equivalent  to  an  adverb,  confers  mo- 
dality on  a  proposition.  '  Error  is  always  in  haste,'  '  Justice 
is  ever  equal,'  'A  perfect  man  ought  always  to  be  conquering 
himself,'  are  examples  of  modal  propositions  in  this  accepta- 
tion of  the  name.  Other  logicians,  however,  have  adopted  a 
different  view,  and  treat  modality  as  consisting  in  the  degree 
of  certainty  or  probability  with  which  a  judgment  is  made  and 
asserted.  Thus,  we  may  say,  '  An  equilateral  triangle  is  neces- 
sarily equiangular,'  'Men  are  generally  trustworthy,'  '  A  falling 
barometer  probably  indicates  a  coming  storm,' '  Aristotle's  lost 
treatises  may  possibly  be  recovered  ; '  and  all  these  assertions 
are  made  with  a  different  degree  of  certainty  or  modality." 
But  this  does  not  affect  the  nature  and  relations  of  the  copula, 
and  if  we  remain  by  the  definition  of  the  predicate,  we  shall 


PROPOSITIONS  OR  JUDGMENTS  115 

find  that  modal  articles  and  terms  simply  modify  attributives, 
verbal  or  adjectival,  and  no  special  significance  should  be  at- 
tached to  them  when  they  do  not  affect  the  quantity  of  the 
proposition. 

Some  logicians  distinguish  propositions  into  True  and  False. 
But  this  has  to  do  with  their  matter  as  valid,  and  not  their  form, 
as  a  mode  of  thinking,  and  as  Logic  is  of  formal  laws  it  is  not 
concerned  with  the  material  truth  or  falsehood  of  propositions. 
A  system  of  pure  and  formal  Logic,  correctly  illustrating  the 
laws  of  thought,  could  be  constructed  upon  materially  false 
propositions  as  well  as  upon  true  ones.  It  is  not  a  science  of 
truth  in  general  ;  but  only  of  the  formal  laws  of  thought.  It 
is,  therefore,  not  concerned  whether  propositions  be  true  or 
false. 

3d.  Ambiguity  of  Propositions. — Judgments  are  rendered 
ambiguous  in  three  ways  :  First,  by  the  ambiguous  use  of  certain 
terms  ;  second,  by  the  inverted  position  of  certain  terms  and 
clauses  ;  and  third,  by  the  double  meaning  of  certain  proposi- 
tions even  when  there  is  no  ambiguity  in  any  of  the  terms  com- 
posing it.  The  first  and  the  third  of  these  influences  affect 
propositions  in  the  same  way,  giving  them  a  double  import, 
in  which  one  of  the  implied  propositions  is  the  complement  of 
the  other.  They  may  be  called  Duplex  propositions  because 
they  are  susceptible  of  analysis  into  two  distinct  judgments. 
Those  due  to  the  second  cause  may  be  called  Inverted  propo- 
sitions. 

1.  Invested  Propositions. —  These  are  of  two  kinds,  accord- 
ing as  the  inversion  is  of  the  subject  and  predicate,  or  of 
some  relative  clause.  In  regard  to  the  first,  an  example,  such 
as  may  frequently  be  found  in  poetry,  is,  "  Full  short  his 
journey  was,"  or,  "  Great  is  Diana  of  the  Ephesians."  In  such 
cases  the  order  of  subject  and  predicate  must  be  reinverted 
before  the  proposition  can  be  dealt  with  logically  according  to 
the  formal  rules  of  conversion  and  reasoning.  In  regard  to 
the  second  class,  the  subject  may  sometimes  be  mistaken  for 
the  predicate  when  it  is  described  by  a  relative  clause  stand- 
ing at  the  end  of  the  sentence  ;  as,  "  No  man  is  honest  who 


110  ELEMENTS  OF  LOCK' 

cheats  liis  neighbor,"  or,  "No  one  is  fit  for  a  king  who  cannot 
rule  himself."  The  real  subjects  in  these  propositions  are, 
"No  one  who  cheats  his  neighbor," and  "  No  one  who  cannot 
rule  himself,"  and  unless  we  keep  this  fact  in  mind  such  in- 
stances would  give  trouble  in  determining  the  Figure  of  h  syllo- 
gism, as  will  appear  when  thai  subject  is  to  be  discussed. 

2.  Duplex  Propositions. — A  duplex  proposition  is  onewhi-h 
is  capable  of  a  double  meaning  and  can  be  analyzed  into  two 
distinct  judgments.  There  are  three  kinds:  Partitive,  Ex- 
clusive, and  Exceptive.  The  chief  characteristic  of  these  prop- 
ositions is,  limi  the  complementary  proposition  implied  by  them 
is  of  the  opposite  quality  of  that  which  is  asserted  in  the  given 
instance.  This  will  be  very  important  to  keep  in  mind,  be- 
cause the  process  of  reasoning  will  be  affected  by  the  question 
whether  one  or  the  other  of  them  is  the  real  one  in  the  thought 
of  the  reasoner,  as  will  be  illustrated. 

(a)  Partitive  Propositions. — These  express  a  part  of  a  whole, 
of  which  the  implied  proposition  is  a  complementary  part, 
and  are  determined  by  the  ambiguous  use  of  the  terms  "All — 
not,"  "  Some"  and  "  Few."  "All — not  "is  often  conceived  as 
the  same  as  "  Not  all,"  and  hence  when  the  proposition  seems 
to  be  universal  it  is  really  particular.  As  an  illustration  we 
have,  "All  metals  are  not  denser  than  water,"  or,  "  All  men  are 
not  red-haired,"  where  we  may  mean  that  "Not  all  metals  are 
denser  than  water,"  and  "  Not  all  men  are  red-haired."  Strictly 
construed  the  original  propositions  are  E  in  form,  but  in  mat- 
ter they  are  either  I  or  O,  with  the  other  of  the  two  implied 
when  one  of  them  is  distinctly  intended.  When  I  say  that 
"Not  all  men  are  red-haired,"  or  "All  men  are  not  red-haired," 
in  the  sense  of  the  former,  I  mean  that  "  Some  men  are  red- 
haired,"  and  that  "  Some  men  are  not  red-haired."  Whichever 
of  the  two  I  have  in  thought,  the  other  is  implied  as  its  com- 
plement. 

Again,  the  term  "  some "  is  subject  to  a  similar  ambiguity, 
denoting  some  but  not  all,  and  some  at  least,  and  it  may  be  alb 
Thus  the  proposition  "Some  metals  are  precious,"  especially 
if,  in  speaking,  the  emphasis  be  upon  the  word  "  some,"  may 


PROPOSITIONS  OR  JUDGMENTS  117 

mean  that  "Some  metals  are  precious,"  and  "Some  metals  are 
not  precious."  This  is  when  the  term  is  equivalent  to  not  all, 
or  only  a  part.  In  such  instances  it  implies  its  complementary 
opposite,  so  that  it  means  I  and  O  at  the  same  time.  If  it  be 
I,  it  implies  O  ;  if  it  be  0,  it  implies  I.  The  strict  and  proper 
import  of  the  term,  however,  when  describing  particular  prop- 
ositions is  that  in  which  it  denotes  "  some,  and  there  may  or 
may  not  be  all."  The  importance  of  this  will  appear  in 
considering  the  matter  of  Opposition.  But  in  actual  reason- 
ing we  must  be  on  the  alert  for  the  ambiguity  to  which  the 
term  is  incident,  and  be  ready  to  detect  the  fallacy  which  it 
may  occasion. 

A  third  proposition  of  a  partitive  and  duplex  nature  is  that 
introduced  by  the  term  "few ;  "  as,  "  Few  cities  are  as  large  as 
Vienna,"  or,  "  Few  men  can  be  President,"  etc.,  in  which  we 
mean  that  "  Most  cities  are  not  as  large  as  Vienna,"  and  "  Most 
men  cannot  be  President."  Such  propositions  imply  a  com- 
plementary opposite,  because  the  term  "  few  "  denotes  some, 
but  not  all,  or  a  few,  but  not  all.  The  expression  "a  few" 
taken  alone  does  not  imply  any  complementary  conception,  but 
is  equivalent  to  the  unambiguous  use  of  the  word  "  some." 
"A  few,"  therefore,  introduces  a  proposition  which  will  be 
either  I  or  O  alone,  unless  "  A  few — not "  be  regarded  as  am- 
biguous like  "  All — not."  But  "  few  "  introduces  a  proposition 
which  has  the  meaning  of  I  and  O  together,  as  the  illustrations 
given  very  clearly  prove.  The  confusion  to  which  such  proposi- 
tions may  give  rise  will  be  seen  in  those  forms  of  reasoning 
where  the  validity  of  the  conclusion  turns  upon  the  question 
whether  they  are  to  be  interpreted  as  I  or  O.  Thus  the  danger 
can  be  illustrated  : 

All  men  are  mortal. 

Few  representatives  of  charity  work  are  men. 
\  Few  representatives  of  charity  work  are  mortal. 

Taking  "  few "  in  its  duplex  import,  the  conclusion  would 
mean  that  "  Most  representatives,  etc.,  are  not  mortal,"  when 
we  know  that  they  are  all  so.     Hence  we  cannot  treat  the 


118  ELEMENTS  OE  LOGIC 

minor  premise  as  a  simple  unambiguous  proposition,  but  must 
interpret  it  as  meaning  both  I  and  O,  in  which  the  conclusion 
would  be  valid  with  I,  where  "  few  "  is  equivalent  to  "  some  " 
(unambiguous),  but  vitiated  with  O,  for  reasons  that  will  ap- 
pear when  discussing  the  doctrine  of  fallacies. 

(b)  Exclusive  Propositions. — They  are  introduced  or  have 
their  meaning  determined  by  such  particles  as  only,  alone,  and 
none  but.  They,  therefore,  limit  the  predicate  to  the  subject, 
and  are  illustrated  by  such  propositions  as  "  Only  Caucasians 
are  white,"  "  Giants  alone  can  be  seven  feet  tall,"  "  Elements 
alone  are  metals,"  "  None  but  honest  men  can  be  trusted,"  etc. 
When  I  say  that  "only  elements  are  metals,"  I  do  not  neces- 
sarily mean  that  "  all  elements  are  metals,"  but  that  the  class 
"  metal"  belongs  exclusively  to  the  class  "  element,"  and  that 
it  can  be  affirmed  of  no  other  class.  Hence  the  meaning  of 
the  proposition  is  either  "  All  metals  are  elements,"  or,  "  All 
compounds  (not-elements)  are  not  metals."  The  first  of  these 
is  what  is  called  the  simple  converse,  and  the  second  may  be 
called  the  complementary  opposite  of  the  exclusive  proposition 
in  question.  It  is  with  one  of  these,  or  the  conception  ex- 
pressed by  one  of  them,  with  which  we  have  to  deal  in  reason- 
ing, or  in  testing  any  case  of  reasoning  involving  an  exclusive 
proposition.  Thus,  "  Only  Caucasians  are  white  "  must  be  re- 
duced either  to  "  All  white  men  are  Caucasians,"  or  to  "  Those 
who  are  not  Caucasians  are  not  white,"  when  testing  the  formal 
process  of  the  syllogism.  The  error  to  which  we  are  liable  in 
using  them  without  considering  their  duplex  meaning  is  illus- 
trated in  the  following  argument : 

Only  elements  are  metals, 
Oxygen  is  an  element. 
.'.  Oxygen  is  a  metal. 

Now  we  know  that  oxygen  along  with  a  number  of  other  sub- 
stances is  not  a  metal.     Or  again,  in  a  better  case  : 

Only  men  are  allowed  to  vote. 

Criminals  are  men. 

.\  Criminals  are  allowed  to  vote. 


PROPOSITIONS  OR  JUDGMENTS  119 

But  we  know  that  criminals  are  not  allowed  to  vote,  and  hence 
to  test  the  character  of  the  reasoning  we  must  use  either  the 
converse  or  the  complementary  of  the  proposition  in  the  major 
premise.     Thus  if  I  say  : 

All  who  are  allowed  to  vote  are  men. 
Criminals  are  men, 

it  will  be  seen  that  I  can  draw  no  conclusions,  for  reasons  to 
be  noted  when  we  study  the  syllogism  ;  and  so  with  the  com- 
plementary proposition  ;  as, 

Those  who  are  not  men  are  not  allowed  to  vote. 
Criminals  are  men. 

There  arc  cases  also  where  there  might  appear  to  be  a  formal 
error  in  the  reasoning,  but  which  is  perfectly  correct  when  we 
consider  either  the  converse  or  the  comphmentary  of  the  ex- 
clusive proposition  expressed.     Thus  : 

Only  elements  are  metals. 

Gold  is  a  metal. 

.*.  Gold  is  an  element. 

This  is  correct,  but  the  reason  for  it  will  appear  again. 

It  is  important  to  say  a  few  words  about  the  quantity  of  ex- 
clusive propositions,  because  the  question  may  be  asked 
whether  they  are  universal  or  particular.  The  answer  cannot 
be  made  without  an  explanation  of  this  peculiarity.  Exclusive 
propositions  are  of  two  forms.  An  illustration  of  the  first  form 
is,  "  Only  citizens  can  vote,"  and  of  the  second  form,  "  Only  some 
men  are  wise."  The  first  may  seem  to  be  universal,  and  the  sec- 
ond particular.  The  second,  which  is  a  proposition  in  I,  is  a 
particular  proposition,  and  differs  from  the  ordinary  instance 
only  in  implying  the  complementary  opposite,  O.  The  first 
form  is  the  most  frequent,  and  has  the  peculiarity  that  the 
terms  "only,"  "alone,"  and  "none  but,"  have  the  effect  of  dis- 
tributing the  predicate  while  the  subject  is  left  undistributed. 
The  meaning  of  distribution  will  have  to  be  ascertained  in  its 
proper  place,  but  we  may  say  regarding  it  at  present,  that  it 


120  ELEMENTS  OF  LOGIC 

denotes  that  the  whole  extension  of  the  concept  distributed  is 
taken  into  account.  When  undistributed,  a  part  of  that  ex- 
tension is  taken  into  account,  and  it  may  or  may  not  be  that 
all  of  it  is  considered.  This  makes  the  exclusive  proposition, 
as  it  stands  in  the  first  form,  a  sort  of  inverted  universal. 
But  it  is  neither  a  universal  nor  a  particular  in  that  form. 
Hence  we  can  determine  its  quantity  only  by  taking  either 
its  simple  converse,  or  its  complementary,  and  in  either  of 
these  cases  the  result  is  a  universal.  Consequently,  the  first 
form  of  the  exclusive  proposition  must  be  treated  in  terms  of 
what  it  implies,  namely,  a  universal  proposition. 

The  second  form  is  the  ordinary  particular  proposition  and 
must  be  treated  accordingly,  except  that  the  signs  "  only," 
"  alone,"  etc.,  do  not  distribute  the  predicate.  They  have  only 
the  effect  of  implying  the  complementary  opposite  proposi- 
tion O. 

There  is  one  class  of  exclusive  propositions  to  which  these 
observations  do  not  apply.  They  are  the  negative  cases,  such 
as  "  Only  bad  men  are  not  wise."  But  as  such  instances  are 
not  frequent  in  practical  reasoning,  it  is  sufficient  to  warn  the 
student  against  applying  the  preceding  principles  to  them. 
They  have  no  complementary  opposite,  because  in  the  com- 
plementary proposition  the  predicate  is  undistributed,  and  we 
cannot  argue  from  the  exclusion  of  one  conception  from  an- 
other to  the  inclusion  of  its  opposite  in  any  particular  con- 
ception. Again,  negative  exclusive  propositions,  from  the  fact 
that  the  subjects  are  not  distributed,  are  in  reality  particular 
propositions  in  O,  and  hence  cannot  be  converted.  No  rules, 
therefore,  can  be  laid  down  respecting  them. 

(c)  Exceptive  Propositions. — These  are  such  as  are  introduced 
or  modified  by  the  terms  All  but,  All  except,  All  save,  etc. 
For  example,  "  All  except  those  under  twenty-one  years  of 
age  are  citizens,"  "  All  the  planets,  except  Venus  and  Mercury, 
are  beyond  the  earth's  orbit."  Such  propositions  appear  to  be 
universal,  and  simple  at  the  same  time.  But  they  really  con- 
sist of  two  particular  propositions  ;  namely,  I  and  O ;  as, 
"  Some  men  are  citizens/'  and  "  Some  men  are  not  citizens,"  or, 


PROPOSITIONS   OR  JU1XJMKNTS 


121 


"  Some  planets  are  beyond  the  earth's  orbit,"  and  "  Some  planets 
are  not  beyond  the  earth's  orbit."  If  the  class  "  men  "  or  the 
class  "  planets  "  were  divided  into  two  species  with  a  name  ac- 
cording to  the  two  portions  indicated  by  the  nature  of  the 
subject,  exceptive  propositions  might  be  resolved  into  two 
universalis,  A  and  E,  instead  of  two  particulars,  land  O.  Thus, 
"  All  who  are  twenty-one  years  of  age  and  over  are  citizens," 
and  "  All  who  are  below  twenty-one  are  not  citizens."  But  in 
either  case  we  have  an  illustration  of  the  complementary  nature 
of  the  two  propositions  developed  from  the  duplexity  of  an  ex- 
ceptive judgment.  This  class,  however,  is  not  so  important  in 
Logic  as  the  two  previous  classes,  because  fallacies  are  less 
frequently  incident  to  the  use  of  them.  We  require  only  to 
observe  the  peculiar  nature  of  the  conception  involved  in  such 
judgments,  aDd  to  be  on  the  alert  for  any  disturbing  influence 
it  is  likely  to  exercise.* 

The  following  outline  is  a  resume  of  the  chapter : 


Propositions 


'  Definitions  ( Indicative. 

Grammatical  J.  Interrogative. 
(  Imperative. 

I  Categorical. 
Grammatico-Logical  <  Conditional. 
(  Disjunctive. 
(  Universal  .<  Affirmative  =  A. 
Quanto-qualitative-^  I  a«B       ♦*■      =  ?' 

Divisions-  1  Particular]  Affirmative  =  I. 

Psychological  j^ytic. 

(  Truistic. 
Miscellaneous  <  Pure. 
(  Modal. 

( Inverted. 
,  Ambiguous  propositions  <  (  Partitive. 

(  Duplex  <  Exclusive. 
( Exceptive. 

*  General  references  on  Propositions  are  the  following:  Mill:  Logic, 
Bk.,  I.,  Chaps.,  IV.,  V.,  and  VI  ;  also  Examination  of  the  Philosophy  of 
Sir  William  Hamilton.  Chap.  XVIII  ;  Hamilton :  Lectures  on  Logic, 
Lect.  XIII.  ;  Venn :  Empirical  Logic,  Chaps.  IX.  and  X.  ;  De  Morgan  : 
Formal  Logic,  Chap.  IV.  ;  Wundt :  Logik,  Dritter  Abschnitt,  Cap. 
II.  ;  Keynes  :   Formal  Logic,  Part  II.,  Chap.  I. 


CHAPTER  VIII. 

THE  RELATION  BETWEEN  SUBJECT  AND  PREDICATE 

I  st.  Nature  of  the  Relation  Between  Subject  and  Pred- 
icate.— We  have  briefly  alluded  to  the  relation  between  sub- 
ject and  predicate  without  explaining  it.  This  was  in  the  case 
of  drawing  a  distinction  between  the  judgments  "  Man  is  a 
biped  "  and  "  Man  is  wise."  But  now  this  relation  must  be  ex- 
amined more  carefully  and  illustrated  by  symbolic  diagrams 
which  will  be  convenient  for  testing  visibly  certain  forms  of 
reasoning  and  inference. 

There  has  been  much  dispute  about  the  nature  of  this  rela- 
tion. One  set  of  logicians  has  claimed,  and  the  other  has 
denied,  that  the  relation  can  be  expressed  in  terms  of  quantity  ; 
in  other  words,  that  the  subject  and  predicate  are  considered 
as  expressing  merely  a  relation  of  quantity.  Thus  the  judg- 
ment "  Men  are  bipeds  "  is  supposed  to  have  its  meaning  indi- 
cated in  the  notion  that  the  subject  expresses  a  number  of 
individuals  not  greater  and  possibly  less  than  the  number 
indicated  by  the  predicate.  This  conception  refers  us  to  the 
matter  already  discussed  ;  namely,  that  of  extension.  In  for- 
mal Logic  it  has  been  customary  to  deal  with  all  propositions 
as  if  we  had  to  take  no  other  relation  into  account,  and  owing 
to  the  peculiarly  different  character  of  such  propositions  as 
"Man  is  wise"  the  correctness  and  accuracy  of  the  general 
practice  has  been  impeached  by  some  writers,  and  more  par- 
ticularly the  mode  of  representing  the  relation  by  geometrical 
figures.  It  is  the  merits  of  this  question  which  we  wish  to  ex- 
amine with  some  care. 

The  student  must  remember  what  has  been  said  about  the  ex- 
tension of  terms  and  the  laws  regarding  the  relation  between 


RELATION  BETWEEN  SUBJECT  AND  PREDICATE    123 

intension  and  extension.  In  the  present  problem  it  has  been 
customary  to  deal  only  with  their  relations  in  extension  and 
to  choose  geometrical  figures  to  represent  them  in  order  to 
show  some  of  the  characteristics  affecting  the  process  of  rea- 
soning. If  propositions  express  a  relation  of  quantity,  of 
equality,  of  more  or  less,  between  terms  it  would  be  natural  to 
represent  them  mathematically.  But  as  the  proposition  "  Man 
is  wise  "  does  not  seem  to  indicate  a  relation  of  quantity  or  ex- 
tension between  subject  and  predicate,  but  a  relation  of  attri- 
bute to  its  substance  ;  and  as  judgments  like  "  Man  is  a  biped  '' 
imply  the  connection  of  attribute  and  substance,  whatever  else 
is  thought  of,  it  has  been  maintained  that  symbols  of  quantity 
representing  the  relation  are  misleading.  This  is  the  problem 
to  be  considered. 

In  examining  the  relation  between  subject  and  predicate,  we 
may  adopt  a  division  of  propositions  or  judgments  which  has 
not  been  mentioned,  but  which  is  based  upon  the  distinction 
between  the  extension,  and  the  intension  of  concepts.  Accord- 
ingly all  judgments  may  be  divided  into  judgments  of  extru- 
sion, and  judgments  of  intention.  An  example  of  the  former 
is,  "  Man  is  a  biped,"  or  "  Horses  are  animals  ; "  of  the  latter 
examples  are,  "  Man  is  wise,"  and  "  Trees  are  tall."  A  good 
way  to  represent  the  difference  is  found  in  the  following  prop- 
ositions, the  first  extensive  and  the  second  intensive  :  "  Man 
is  a  mortal," and  "Man  is  mortal."  The  great  difference  be- 
tween them  is  remarked  in  the  nature  of  the  predicate  and  its 
relation  to  the  subject.  In  all  propositions  the  predicate  is 
either  substantive  or  attributive.  In  judgments  of  extension 
the  predicate  is  substantive  ;  in  judgments  of  intension  it  is 
attributive.  As  we  have  already  explained,  such  propositions 
as  "John  struck  James,"  or  "  The  king  rules  his  subjects,"  the 
verb  expresses  a  function  or  attribute  of  the  subject,  or  an 
attribute  that  is  attributive,  and  hence  the  presence  of  a  sub- 
stantive object  does  not  affect  the  attributive  nature  of  the 
predicate,  or  the  intensive  nature  of  the  judgment.  But  with 
this  uniform  relation  of  a  substantive  predicate  to  extensive 
and  of  an  attributive  predicate  to  intensive  judgments,  we  may 


124  ELEMENTS  OF  LOGIC 

remark  another  important  distinction  which  is  its  corollary. 
In  the  extensive  proposition  the  subject  is  contained  or  compre- 
hended in  the  predicate  or  excluded  from  it  ;  in  the  intensive 
proposition  the  predicate  is  contained  or  comprehended  in  the 
subject  or  excluded  from  it.  The  mode  of  comprehension,  how- 
ever, is  distinct  in  each  case.  In  the  former  it  may  be  called 
that  of  inclusion  or  exclusion  ;  in  the  latter  that  of  inhesion  or 
non-inhesion.  By  inclusion  of  the  subject  in  the  predicate  I 
mean  that  it  is  contained  as  a  species  in  the  genus,  or  as  one 
class  in  another  class  term  of  equal  or  greater  extension.  The 
exclusive  proposition  is  no  exception  to  this,  because  we  found 
that  its  logical  meaning  was  the  converse  of  its  grammatical 
form.  Hence  the  predicate  of  the  extensive  judgment  is  a 
class  concept.  By  the  inhesion  of  the  j)redicate  in  the  subject 
I  mean  that  it  is  contained,  or  inheres,  in  the  subject  as  an 
attribute  in  a  substance,  or  rather  expresses  that  relation.  In 
the  extensive  judgment,  therefore,  the  number  of  individuals 
denoted  by  the  subject  can  never  be  greater  than  the  number 
denoted  by  the  predicate.  This  establishes  a  relation  of  quan- 
tity between  them,  and  hence  they  may  be  called  quantitative 
judgments.  The  relation  being  quantitative  can  be  represented 
in  some  mathematical  way,  if  not  to  indicate  the  nature  of  it, 
certainly  to  indicate  an  accident  quite  uniform  with  the  essen- 
tial qualities  and  proportionally  variable  with  them.  In  the 
intensive  judgment  the  predicate  is  an  attribute  or  quality  of 
the  subject,  and  is  comprehended  in  it  rather  than  the  re- 
verse, as  in  the  extensive  proposition.  Intensive  may,  there- 
fore, be  called  qualitative  judgments,  because  the  connec- 
tion between  subject  and  predicate  is  a  relation  of  qua/ it;/.  It 
is,  therefore,  a  question  whether  they  can  be  represented  or 
symbolized  by  any  figures  expressing  relations  of  quantity. 

The  answer  to  this  question  will  be  found  in  the  fact  to  be 
shown  that  both  kinds  of  judgment  can  be  conceived  in  both 
a  quantitative  and  a  qualitative  form  at  the  same  time,  and 
qualitative  in  a  double  sense.  If  this  can  be  proved,  the  sym- 
bolization  of  one  will  be  that  of  the  other  also.  We  proceed, 
therefore,  to  the  examination  of  this  question. 


RELATION  BETWEEN  SUBJECT  AND  PREDICATE    125 

First,  in  extensive  judgments,  although  the  predicate  is  a 
substantive  and  class  concept,  it  connotes  certain  attributes 
which  belong  to  the  subject  in  the  same  way  as  in  the  inten- 
sive judgment.  Thus,  when  I  say  "Man  is  a  biped,"  I  not 
only  mean  that  the  class  of  individuals  or  species  "  man  "  is 
included  in  the  equal  or  larger  class  "  biped,"  but  I  also  mean 
that  "  man  "  is  characterized  by  the  quality  of  two-footedness, 
common  to  the  whole  class  of  bipeds.  I  therefore  affirm  or 
imply  this  attribute  of  him,  and  the  extensive  judgment  be- 
comes at  the  same  time  an  intensive  one. 

Second,  in  intensive  judgments,  although  the  predicate  is 
an  attribute  term,  it  is  generally  assumed  that  it  cannot  stand 
alone  in  thought,  but  must  qualify  some  substantive.  Thus, 
when  I  say  "  Man  is  wise,"  I  not  only  mean  that  wisdom  is  an 
attribute  of  the  subject,  but  I  equally  mean  that  he  is  a  wise 
something.  I  do  not  mean  that  "  Man  is  a  wise  man,"  for  this 
is  tautological.  But  I  mean  that  he  is  a  "  wise  creature,"  in 
which  case  I  have  a  substantive  predicate,  as  implied  in  the 
simple  intensive  form,  and  the  intensive  judgment  becomes  an 
extensive  one  at  the  same  time.  It  is  true  that  we  do  not  or- 
dinarily, perhaps  never,  think  of  this  class  relation  in  such  a 
judgment  as  "  Man  is  wise,"  but  the  fact  that  the  extensive 
conception  of  it  coincides  and  is  perfectly  compatible  with 
the  intensive  conception,  is  sufficient  to  give  it  that  double 
logical  construction,  as  in  the  case  of  the  extensive  judgments. 
This  is  especially  the  fact  when  we  reflect  that  in  the  extensive 
judgments  we  may  not  ordinarily  represent  to  our  thought 
the  attributive  relation  between  subject  and  predicate  any 
more  than  in  the  intensive  judgment  we  represent  the  class 
relation,  and  yet  no  one  questions  that  the  attributive  relation 
is  implied  in  the  extensive  judgment.  The  extensive  relation 
is  equally  involved,  or  implied  in  the  intensive  judgment,  al- 
though it  may  not  be  thought  of.  At  any  rate,  it  is  possible 
to  represent  it  so,  in  perfect  compatibility  with  the  attribu- 
tive relation  more  particularly  expressed  by  it.  This  will  ena- 
ble us  to  represent  the  relation  quantitatively  or  mathemati- 
cally as  in  the  extensive  judgment. 


126  ELEMENTS  OF  LOGIC 

Another  fact  sustains  the  same  conclusions.  When  I  say 
"  Man  is  wise,"  there  is  nothing  in  the  nature  of  this  form  of 
statement  to  prevent  my  affirming  wisdom  of  other  beings  as 
well.  Take  the  proposition  "  Trees  are  tall,"  and  we  may  also 
say,  "Houses  are  tall."  "Tallness"  is  not  exclusively  an  attri- 
bute of  "  trees,"  and  hence  "  trees  "  belong  to  a  larger  class  of 
objects  having  the  same  quality,  "tallness,"  as  "man"  belongs 
to  a  larger  class  of  beings  having  the  quality  "  two-footed- 
ness,"  expressed  in  the  proposition  "  Man  is  a  biped."  Hence, 
so  far  as  form  of  statement  is  concerned,  the  judgment  "  Man 
is  wise,"  may  admit  that  other  beings  are  "  wise  "  as  well,  and 
as  long  as  this  is  the  case,  formally,  the  judgment  is  extensive 
as  well  as  intensive. 

It  may  be  important  to  consider  the  relation  between  the 
judgment  "  Man  is  a  wise  creature  "  and  a  definition.  I  have  said 
that  the  predicate  "  wise,"  in  the  simple  intensive  proposition, 
may  be  affirmed  of  other  individuals  and  species  than  the  sub- 
ject, and  that  this  constitutes  a  significant  resemblance  to  exten- 
sive propositions,  because  their  fundamental  characteristic  is 
precisely  this  fact,  that  it  is  affirmable  of  other  individuals  be- 
sides the  given  subject.  But  this  is  not  the  case  with  a  defini- 
tion. In  a  definition  the  subject  and  predicate  are  identical  in 
extension,  and  convertible  with  each  other.  Thus,  if  I  define 
man  to  be  a  rational  animal,  I  can  as  well  say  that  "  Rational 
animals  are  men."  This  is  merely  because  I  regard  the  property 
"  rational  "  as  belonging  exclusively  to  man,  and  so  make  it  the 
differentia,  while  the  word  "  animal  "  refers  to  the  conferentia.  It 
is,  therefore,  the  total  predicate  which  is  identical  with  the  sub- 
ject, while  the  generic  term,  with  its  conferentia,  indicates  an 
extensive  relation  numerically  greater  than  the  subject,  and  it 
is  only  the  differentia  that  can  make  the  total  equal  to  the 
subject.  Now  it  is  to  be  remarked  that  the  conversion  of  a 
simple  intensive  judgment,  such  as  "  Man  is  wise,"  into  its 
corresponding  extensive  judgment,  such  as  "  Man  is  a  wise 
creature,"  looks  very  much  like  a  definition,  and  hence  the 
wisdom  might  not  be  predicable  of  anything  else  than  the 
subject.     It  might  bo  argued  that   this  is  the  possible  case 


RELATION  BETWEEN  SUBJECT  AND  PREDICATE    127 

with  all  intensive  judgments,  and  if  so  their  resemblance  to  ex- 
tensive judgments  is  modified.  But  the  reply  to  this  is  very 
clear. 

In  the  first  place,  extensive  judgments  do  not  require  that 
the  predicate  he  greater  in  extension  than  the  subject.  It  may 
be  equal  to  it  and  supply  all  the  conditions  necessary.  A  defi- 
nition, therefore,  may  be  an  extensive  judgment,  and  can  always 
be  treated  so.  Farther,  it  is  important  to  remark  that,  formally, 
a  definition  has  to  be  treated  as  all  other  propositions,  and  it 
is  only  materially,  that  is,  when  we  consider  the  modifying 
attributive  as  a  differentia,  that  we  can  treat  its  predicate  as 
convertible  with  the  subject.  In  the  second  place,  the  attri- 
bute "wise"  in  this  particular  judgment,  and  the  qualifying 
term  in  any  other  proposition,  may  indicate  either  the  confer- 
entia  or  the  accidentia,  and  in  either  case  involve  a  predicate  of 
broader  extension  than  the  subject,  when  a  substantive  is  modi- 
fied by  them.  It  is  only  when  the  modifier  expresses  the  differ- 
entia that  the  predicate  can  ever  be  equal  in  extension  to  the 
subject,  assuming  it  to  qualify  a  substantive.  Hence  all  judg- 
ments which  are  not  definitions  represent  predicates  of  greater 
extension  than  the  subject,  and  as  even  definitions  cannot  be 
formally  distinguished  from  them  as  such,  they  must  be  treated 
logically  in  the  same  way,  although  when  materially  known  to 
be  definitions  we  may  consider  their  extension  as  reduced  in 
reality  to  that  of  the  subject.  But  even  this  does  not  prevent 
them,  as  we  have  shown,  from  being  extensive  judgments,  and 
therefore,  whenever  an  intensive  proposition  is  assumed  to 
imply  a  substantive  element  in  the  predicate,  it  possesses 
quantitative  properties  identical  in  character  with  those  of 
the  so  -  called  extensive  judgment  and  may  be  represented 
accordingly.  The  figured  symbols  representing  the  quanti- 
ty, or  relations  of  quantity  in  extension  between  subject  and 
predicate,  shall  be  illustrated  presently,  and  as  soon  as  another 
interesting  feature  of  both  kinds  of  judgment  has  been 
considered. 

What  we  have  considered  up  to  this  point  in  the  two  forms 
of  proposition  is  quantity  of  extension,  expressed  or  implied. 


12S  ELEMENTS  OF  LOGIC 

What  we  have  still  to  consider  is  quantity  of intension  ;  for  this 
is  as  marked  a  characteristic  of  judgments  as  any  other  prop- 
erty, although  it  is  not  usual  to  regard  it  in  symbolic  Logic. 
But  take  the  simple  intensive  judgment,  "  Man  is  wise."  We 
say  the  predicate  is  here  contained  in  the  subject,  or  denotes 
a  property  belonging  to  it.  Not  only  is  it  to  be  remarked 
that  the  same  predicate  may  belong  to  other  subjects  also,  but 
this  one  is  not  the  only  property  of  the  subject.  As  in  the  ex- 
tensive proposition  the  subject  can  never  represent  a  greater 
number  of  individuals  than  the  predicate,  excepting  in  nega- 
tive propositions  as  explained,  so  in  the  intensive  proposition 
the  predicate  can  never  represent  a  greater  number  of  attri- 
butes than  the  subject,  except  in  negative  propositions,  as  be- 
fore. The  quantity  of  intension,  therefore,  of  the  subject  in 
intensive  judgments  must  be  equal  to,  or  greater  than,  that  of 
the  predicate.  Consequently  the  mode  of  symbolizing  the 
mathematical  relation  would  be  the  reverse  of  that  in  the  ex- 
tensive proposition ;  that  is,  with  the  same  figures,  but  with  a 
reversed  position  for  the  signs  of  the  subject  and  predicate. 
Now,  as  we  have  shown  that  the  extensive  proposition  has  its 
intensive  interpretation ;  thus,  "  Man  is  a  biped  "  equals  "  Man 
is  two-footed,"  denoting  qualitatively  what  is  implied  by  the 
extensive  form,  the  quantity  of  intension  between  subject  and 
predicate  is  the  reverse  of  the  quantity  of  extension  between 
the  same  terms,  and  hence  the  mode  of  representing  it  sym- 
bolically will  be  the  reverse  again.  This  we  proceed  now  to 
illustrate  in  full.  But  we  must  first  explain  how  it  is  done,  and 
shall  then  represent  the  quantity  of  extension,  'as  if  it  applied 
only  to  extensive  judgments.  Afterward  we  can  extend  the 
principle  to  intensive  propositions. 

We  should,  perhaps,  remark  a  connection  between  subject 
and  predicate  which  is  in  some  cases  different  from  the  two 
we  have  discussed,  and  which  would  be  considered  preferable 
to  them  by  certain  schools  of  thought  known  as  Empiricists 
and  Positivists,  or  such  as  oppose  all  Metaphysics.  Instead 
of  supposing  that  subject  and  predicate  expressed  real  objects, 
or  things  in  which  all  predicates  affirmed  inhere  as  qualities  or 


RELATION  BETWEEN  SUBJECT  AND  PREDICATE    129 

attributes,  they  would  say  that  judgments  expressed  the  connec- 
tion of  coexistence  or  sequence  between  subject  and  predicate. 
This  view  would  get  rid  of  the  necessity  of  supposing  the 
subject  to  always  express  a  substantive,  and  the  predicate  a 
substantive  or  attributive  conception,  the  latter  distinctly  indi- 
cating the  inhesion  of  a  quality  in  a  substance,  and  the  former 
implying  it  while  affirming  a  class-whole  to  which  the  subject 
belonged  as  a  numerical  part.  In  this  way  they  would  not 
mean  to  imply  any  necessarily  metaphysical  connection,  but 
only  one  of  coexistence  or  sequence  between  the  two  terms. 
Thus  in  the  proposition  "Man  wears  clothing,"  or  "Man  is 
a  clothed  being,"  we  mean,  they  would  say,  that  the  essential 
qualities  or  characteristics  of  man  are  accompanied  by  the  ac- 
cidental one  of  wearing  clothing,  not  that  a  being  has  this  as 
an  attribute.  This  is  only  to  say  that  certain  facts  or  phe- 
nomena, say  bipedality,  bimanousness,  rationality,  etc.,  are  ac- 
companied by  the  other  quality  of  being  clothed.  The  same 
thing,  perhaps,  could  be  said  of  any  proposition,  such  as 
"  Gold  is  yellow,"  "  Iron  is  hard,"  when  we  suppose  that 
"  gold,"  and  "  iron "  are  names  for  certain  qualities,  among 
which  "  yellow"  may  be  found  in  one  case  and  "  hardness  "  in 
the  other. 

It  is  true  that  many  propositions  seem  to  express,  or  to  be 
resolvable  into,  this  kind  of  connection.  But  it  is  not  opposed 
to  the  kinds  of  connection  we  have  previously  investigated, 
and  may  be  said  to  be  always  coincident  with  them.  Hence, 
while  we  admit  that  such  a  connection  is  most  apparent  in 
many  cases,  it  does  not  exclude  the  idea  that  the  relation  be- 
tween subject  and  predicate  is  that  of  a  subject  and  attribute, 
in  which  we  include,  but  may  not  expressly  think  of,  the  rela- 
tion of  mere  concomitance  or  non-concomitance.  Besides,  in 
extensive  judgments  this  relation  of  mere  connection  by  co- 
incidence or  sequence  is  not  easy  to  imagine,  unless  we  resolve 
the  predicate  into  its  attributive  meaning.  But  it  is  just  as 
•  easy  to  conceive,  and  more  suitable  to  the  traditional  forms  of 
logical  discussion  to  admit,  that  we  think  of  other  relations 
than  of  mere  coincidence  and  sequence.     Judgments  of  exten- 


130  ELEMENTS  OF  LOGIC 

sion  mean  to  indicate,  when  conceived  attributively,  as  they 
in  reality  always  are,  the  conferential  qualities  of  the  subject, 
and  to  ignore  the  differential.  But  judgments  of  intension 
intend  to  express  the  inhesion  of  a  quality  in  the  subject  with- 
out distinction  of  essential  or  accidental.  These  connections 
are  simultaneous  with  that  of  mere  coincidence  or  sequence, 
and  hence  no  important  end  is  served  by  a  controversy  about 
the  question. 

In  order  to  represent  the  relation  between  subject  and 
predicate,  mathematically,  Euler  chose  circles  whose  area  could 
correspond  to  the  suppositions  already  made  about  the  equal 
or  greater  extension  of  the  predicate  as  compared  with  the 
subject.  As  we  have  said,  we  shall  first  limit  the  symbolic 
representation  to  judgments  of  extension,  because  there  can 
be  no  question  about  their  quantitative  nature  and  their  math- 
ematical representation  accordingly.  All  the  figures  we  shall 
employ  may  apply  to  the  extensive  proposition,  "  Metals  are 
substances."  We  shall  use  the  letter  S  to  denote  the  subject, 
and  the  letter  P  to  denote  the  predicate.*  Hence  Swill  stand 
for  "  metals  "  and  P  for  "  substances."  Now  as  the  extension  of 
"  metals  "  cannot  be  greater  than  that  of  "  substances,"  the 
area  of  the  circle  representing  it  must  not  be  greater  but  may 
be  smaller.  Hence  proposition  A  may  be  represented  in  the 
following  manner : 


Fig.  4.  Fig.  5. 

Proposition   I,   which   would   be    "  Some   metals   are   sub- 

*  We  must  not  confuse  this  with  the  later  use  of  the  same  symbols  to 
indicate  the  minor  and  major  terms  of  the  syllogism.  They  are  respect- 
ively the  subject  and  predicate  of  the  conclusion,  but  are  not  always 
such  in  the  premis  s 


RELATION  BETWEEN  SUBJECT  AND  PREDICATE    131 

stances,"  will  be  represented  in  the  following  manner,  to  be  ex- 
plained again  : 


Fig.  6. 


Fig.  7. 


Fig.  8. 


Fig.  9. 


Proposition  E,  "  No  metals  are  substances,"  would  be  repre- 
sented in  only  one  form,  as  follows  : 


Fig.  10. 


Proposition  O,  which  would  be  "  Some  metals  are  not  sub- 
stances," would  require  three  distinct  figures  for  its  symboli- 
zation,  as  follows : 


Fig.  11. 


Fig.   12. 


Fig.  13. 


In  Fig.  4  the  two  circles  are  supposed  to  coincide  and  make 
one,  representing  an   equal    extension  between   subject    ;iud 


132  ELEMENTS  OF  LOGIC 

predicate.  This,  as  remarked,  is  true  of  definitions,  and 
might  be  true  of  all  other  propositions  in  A,  so  far  we  know 
positively  from  the  assertion.  But  the  extension  of  the  predi- 
cate may  be  greater,  as  we  happen  to  know  it  is  in  many  cases, 
and  hence  for  that  conception  Fig.  5  has  to  be  employed.  It 
means  that  the  area  or  number  of  individuals  denoted  by 
"  metals  "  is  greater  than  that  denoted  by  "  substances,"  and 
it  may  always  be  so  in  A  propositions,  so  far  as  we  can  deter- 
mine from  the  form  alone. 

In  proposition  I,  represented  by  Figs.  6,  7,  8,  and  9,  it  is  in- 
teresting to  remark  that  two  of  them  are  identical  with  Figs. 

4  and  5,  and  a  third  differs  from  Fig.  5  only  in  the  position 
of  the  letters  S  and  P.  But  this  resemblance  is  due  to  the 
character  of  particular  propositions.  We  have  remarked  that 
the  sign  "  some  "  properly  denotes  in  Logic  a  part,  and  it  may 
or  may  not  be  all,  and  hence  Figs.  6  and  7  very  clearly  indicate 
this  possibility.  For  if  "  All  S  is  P,"  it  is  evident  that  "  Some  S 
is  P,"  although  the  former  does  not  follow  from  the  latter.  But 
the  conditions  of  a  particular  proposition  being  what  they  are, 
it  is  possible,  so  far  as  the  statement  is  concerned,  that  "  All  S 
is  P,"  when  "  Some  S  is  P."  Figs.  6  and  7  provide  for  this 
possibility.  In  Fig.  8  some  portions  of  S  and  P  are  excluded 
from  each  other,  and  hence  when  it  is  compared  with  Fig.  11, 
it  is  found  to  represent  both  I  and  O,  and  hence  it  might  be 
taken  to  symbolize  the  duplex  proposition,  where  "  some  " 
implies  its  complementary  opposite.  But  it  does  symbolize  I, 
whether  we  regard  it  as  an  ambiguous  representation  or  not. 
Fig.  9  has  no  ambiguity  about  it,  if  we  regard  carefully  the 
relation  expressed  by  the  position  of  the  letters  S  and  P. 
But  it  has  the  fault  of  not  admitting  the  possibility  that  "All 

5  is  P,"  at  the  same  time,  which  is  one  contingency  in  proposi- 
tion I.  So  far  as  we  know  from  proposition  I,  proposition  A 
is  also  true,  and  Fig.  9  does  not  indicate  this. 

We  must  remark,  before  passing  to  the  negative  proposi- 
tions, that  affirmative  judgments  must  express  inclusion.  A 
must  express  total,  and  I  partial  inclusion  at  least.  But  when 
we  come  to  negative  propositions  the  relation  expressed  must 


RELATION  BETWEEN  SUBJECT  AND  PREDICATE    133 

be  one  of  exclusion  :  E  must  express  total,  and  0  partial  ex- 
clusion at  least.  Hence  in  Fig.  10  we  have  the  only  pos- 
sible symbol  of  E.  The  two  excluding  circles  denote  that 
subject  and  predicate  are  not  connected  in  a  given  respect, 
and  hence  no  part  of  the  extension  of  one  can  be  included  in 
that  of  the  other. 

In  proposition  O  the  exclusion  must  be  at  least  partial. 
Fig.  11  represents  it,  and  is  like  Fig.  8  for  I,  although  differ- 
ent arcs  or  portions  of  the  circles  must  be  chosen  to  represent 
the  exclusion,  as  compared  with  the  arcs  in  Fig.  8  to  represent 
inclusion.  Fig.  11,  however,  does  not  indicate  the  possibility 
that  E  may  be  true,  which  is  the  case,  so  far  as  we  know  from 
the  proposition.  Hence  for  the  same  reason  that  Fig.  6  may 
represent  I,  although  also  the  symbol  of  A,  Fig.  13  may  rep- 
resent O,  although  the  symbol  of  E.  Fig.  12  explains  itself 
as  indicating  that  some  of  the  circle  S  is  not  included  in  the 
circle  P  ;  but  it  is  defective  in  not  admitting  the  possibility  of 
E  at  the  same  time.  What  is  desirable  in  all  this  symboliza- 
tion  is  that  the  figures  shall  properly  represent  the  differences 
between  A,  E,  I,  and  O,  and  at  the  same  time  represent  the 
possibility  that  A  is  true  when  I  is,  and  that  E  is  true  when  O 
is.  We  desire  also,  at  the  same  time,  to  represent  the  equal 
possibility  that  A  shall  not  be  true  when  I  is,  and  E  when  O 
is.  In  other  words,  we  require  a  symbol  which  will  represent 
our  entire  ignorance  as  to  whether  there  is  total  or  only  par- 
tial inclusion  implied  when  I  is  affirmed,  or  whether  there  is 
total  or  only  partial  exclusion  when  O  is  affirmed.  If  this  be 
possible  the  number  of  figures  or  symbols  might  be  reduced. 
As  it  is  at  present  we  escape  confusion  only  by  carefully 
observing  the  relation  between  the  various  positions  of  S 
and  P. 

Ueberweg  has  a  representation  which  will  simplify  matters 
very  much.  He  reduces  them  all  to  four  figures,  representing 
respectively  the  propositions  A,  E,  I,  and  O.  He  employs  a 
system  of  dotted  hues  in  order  to  express  the  various  possi- 
bilities involved  in  the  expressed  relation  of  subject  and  pred- 
icate, but  neither  affirmed  nor  denied  by  the  form  of  the  judg- 


134 


ELEMENTS  OF  LOGIC 


nient.  Thus  in  proposition  A,  so  far  as  we  know,  the  exten- 
sion of  the  predicate  may  be  equal  to  or  greater  than  that  of 
the  subject.  The  form  of  the  proposition  does  not  say  which 
it  is.  We  only  know  that  it  cannot  be  less.  Fig.  4  implies 
that  it  is  equal  when  it  may  be  greater,  and  Fig.  5  implies 
that  it  is  greater  when  it  may  be  equal  to  that  of  the  subject. 
Hence  not  only  the  incompleteness  of  the  symbol  in  each  case, 
but  also  the  liability  to  confusion  with  those  for  other  propo- 
sitions. If,  therefore,  we  can  find  symbols  quite  distinct  from 
each  other,  and  yet  expressing  all  the  possibilities  of  the  propo- 
sitions, we  can  greatly  simplify  the  problem.  This  Ueberweg 
has  done  in  the  following  manner,  Fig.  14  standing  for  propo- 
sition A,  Fig.  15  for  E,  Fig.  16  for  I,  and  Fig.  17  for  O  : 


Fig.  14. 


Fig.  15. 


Fig.  16. 


Fig.  17. 


In  Fig.  14,  which  we  see  is  quite  distinct  in  form  from  all 
others,  the  dotted  line  means  that  so  far  as  we  can  tell  from 
the  proposition  A,  "  All  S  is  P,"  the  extension  of  P  may  be 
either  equal  to  or  greater  than  S.  Fig.  15  leaves  no  room  for 
doubt.  It  always  expresses  total  exclusion  and  nothing  else. 
Fig.  16  makes  it  unknown  whether  there  be  a  difference  of 
extension  between  subject  and  predicate,  and  yet  allows  the 
possibility  of  A  being  true  when  I  is  true.  That  is,  it  is  stated 
and  indicated  by  the  undotted  lines  that  "  Some  S  is  P,"  while 
it  may  also  be  true  that  "  All  S  is  P,"  as  is  apparent,  if  S  be 
exhausted  within  the  area  of  the  undotted  lines.     It  is  equally 


RELATION  BETWEEN  SUBJECT  AND  PREDICATE    135 

possible  that  0  be  true  if  S  includes  the  area  of  the  dotted 
lines.  The  same  is  true  of  the  extension  of  P.  It  must  be 
equal  to  or  greater  than  S,  if  proposition  A  be  possible,  and 
must  express  some  exclusion  if  O  be  possible.  The  doubt  in 
both  cases  is  expressed  by  the  dotted  line. 

Fig.  17  is  much  more  complicated.  We  have  to  express 
the  partial  exclusion  of  S  and  P,  the  possibility  of  their  total 
exclusion  ;  that  is,  proposition  E  ;  the  possibility  of  their  par- 
tial  inclusion  ;  that  is,  proposition  I ;  and  the  possibility  of 
the  total  inclusion  of  P  in  S,  as  the  larger  dotted  circle  im- 
plies, and  all  these  at  the  same  time.  A  little  observation  will 
show  that  this  has  been  done.  Thus  if  S  contain  only  what  is 
represented  by  the  undotted  curved  line  and  the  straight  dot- 
ted line,  O  is  symbolized  and  E  is  also  possible.  But  if  S  con- 
tains the  whole  circle  represented  by  the  dotted  and  undotted 
arcs,  O  is  true  and  I  is  possible.  On  the  other  hand,  suppos- 
ing that  S  has  or  can  have  a  larger  extension  than  P,  the 
larger  dotted  circle  represents  again  the  possibility  of  both  O 
and  I.  Taken  altogether  these  four  symbols  are  the  only  com- 
plete ones,  and  are  the  only  representations  which  are  not  lia- 
ble to  confusion  with  each  other. 

An  important  observation  in  the  quantitative  relations  of 
subject  and  predicate  is,  that  in  negative  propositions,  E  and 
O,  no  comparison  of  equal,  greater,  or  less  can  really  be  made. 
When  subject  and  predicate  exclude  each  other,  we  do  not  in- 
dicate anything  about  the  relative  extension  or  number  of  in- 
dividuals denoted  by  them,  and  hence  in  negative  proposi- 
tions relations  of  extension  are  not  commensurable.  As  a 
consequence  of  this,  negative  propositions  either  compare  spe- 
cies which  always  exclude  each  other,  or  compare  genus  and 
species  in  the  inverted  order  of  universal  affirmative  judgments. 
E  compares  co-ordinate  species,  which  are  rej>resented  by 
total  exclusion  because  of  their  distinct  differentia.  O  either 
compares  co-ordinate  species  and  admits  the  possibility  of  E, 
or  total  exclusion  in  regard  to  the  same  species,  or  it  com- 
pares a  genus  with  a  species,  taking  the  genus  for  the  subject 
and  the  species  for  the  predicate.     Thus,  under  the  genus 


136  ELEMENTS  OF  LOGIC 

vertebrate,  we  can  say,  "No  men  are  horses,"  =  E,  or  "Some 
men  are  not  horses  "  =  O,  with  the  possibility  of  E  ;  or  again, 
"Some  vertebrates  are  not  men"  =  O.  But  in  the  last  case 
it  is  not  implied  that  the  extension  of  "  vertebrates  "  can  pos- 
sibly be  greater  than  that  of  the  predicate,  "  men."  Hence,  so 
far  as  the  form  of  judgment  is  concerned,  no  definite  com- 
parison in  the  quantity  of  extension  can  be  made  between  sub- 
ject and  predicate  in  negative  judgments. 

It  is  otherwise  in  affirmative  propositions,  because  they 
express  a  relation  of  inclusion,  and  hence  always  a  relation 
between  genus  and  species,  and  never  between  species  and 
sj)ecies  ;  as,  "  All  men  are  vertebrates  "  and  "  Some  men  are 
negroes." 

Thus  far  we  have  symbolized  only  extensive  judgments,  and 
it  remains  to  see  whether  a  similar  representation  can  be  em- 
ployed for  judgments  of  intension.  It  is  a  very  simple  matter 
to  solve  this  problem.  We  have  only  to  recall  the  previous 
reduction  of  intensive  propositions  to  the  extensive  form,  in- 
volving quantitative  relations  as  well  as  qualitative  between 
subject  and  predicate,  in  order  to  see  that  the  symbolization 
we  have  already  employed  will  apply  equally  to  intensive  judg- 
ments. Hence  the  same  figures  will  represent  the  quantitative 
relations  of  extension  in  such  propositions  as  "  Men  are  wise  " 
and  "Man  is  mortal,"  as  in  "Men  are  bipeds,"  or  "Trees  are 
vegetables."  At  any  rate,  such  a  representation  is  very  con- 
venient for  testing  certain  forms  of  reasoning,  and  is  not  in- 
compatible with  the  relation  of  intension  expressed  by  such 
judgments.  But  assuming  that  all  judgments  are  both  quan- 
titative and  qualitative,  we  may  represent  the  quantitative 
without  interfering  with  the  qualitative  relation,  and  this  is  all 
that  the  symbols  of  Euler  are  intended  to  express.  They  com- 
pare subject  and  predicate  only  in  the  quantity  of  their  exten- 
sion, marking  either  their  inclusion  or  exclusion. 

But  when  we  come  to  compare  subject  and  predicate  in  re- 
spect of  their  quantity  of  intension,  the  matter  is  somewhat 
different.  Here  we  mark  the  relation  of  inhesion  or  non-in- 
hesion, and  it  is  a  question  whether  we  can  symbolize  it  in  any 


RELATION  BETWEEN  SUBJECT  AND  PREDICATE    137 

mathematical  manner  or  not.  But  since  we  have  established 
the  fact  that  the  quantity  of  extension  in  the  relation  between 
subject  arid  rjredicate  is  the  direct  reverse  in  the  relation  ex- 
pressed by  the  quantity  of  intension,  we  may  simply  reverse 
the  positions  of  the  letter  S  and  P  in  order  to  symbolize  the 
mathematical  relation  in  the  intensive  judgment  as  such.  In 
the  proposition  "Man  is  wise,"  we  have  seen  that  the  predicate 
can  never  be  greater  in  intension,  or  quantity  of  intension, 
than  the  subject,  and  that  it  is  contained  in  the  subject.  This 
quantitative  relation  is  very  clearly  represented  by  Figs.  4  and 
5,  only  in  Fig.  5  the  letter  S  will  be  placed  in  the  larger,  and 
the  letter  P  in  the  smaller  circle,  as  in  Fig.  9.  As  Fig.  5  stands 
S  is  contained  in  P,  as  representing  quantity  of  extension,  or 
the  comprehension  of  the  subject  in  the  predicate.  But  as  the 
relation  must  be  reversed  for  the  quantity  of  intension,  or  the 
comprehension  of  the  predicate  in  the  subject,  P  must  be  in- 
cluded in  S.  This  is  the  case  for  the  proposition  A,  which 
Figs.  4  and  5  represent.  For  the  quantity  of  intension  in 
propositions  E,  I,  and  O,  the  same  figures  will  serve  as  for  the 
quantity  of  extension,  only  we  must,  as  in  proposition  A,  re- 
verse the  positions  of  the  letters  S  and  P.  Of  course,  in  Figs. 
4,  6,  8,  10,  11,  and  13,  this  reversal  of  S  and  P  is  not  neces- 
sary. We  require  only  to  keep  in  mind  the  reversed  order  of 
inclusion  as  compared  with  extensive  judgments.  It  is  to  be 
remarked,  however,  that  in  negative  propositions  the  quantity 
of  intension  is  no  more  determinate  than  the  quantity  of  ex- 
tension, relatively  considered,  as  the  same  principles  are  ap- 
plicable here  as  there.  In  the  Ueberweg  scheme  we  require 
to  reverse  the  position  of  the  letters  S  and  P  only  in  Fig.  14, 
and  understand  the  reversed  relation  in  the  others,  although 
in  all  cases  no  harm  will  be  done  by  actually  changing  them  in 
order  to  mark  the  contrast  between  quantity  of  intension  and 
quantity  of  extension. 

It  will  be  evident  that  the  scheme  for  the  quantity  of  inten- 
sion will  apply  also  to  judgments  of  extension,  so  far  as  they 
can  be  reduced,  as  we  have  shown,  to  judgments  of  intension. 
"  All  men  are  bipeds,"  conceived   as  meaning  "  All  men  are 


138  ELEMENTS  OF  LOGIC 

two-footed,"  can  be  represented  in  the  same  way  as  "All  men 
are  wise."  Consequently  we  not  only  have  the  same  symboli- 
zation  for  the  two  kinds  of  judgment,  but  a  double  system 
according  as  each  judgment  is  considered  in  the  quantity  of 
its  extension  or  the  quantity  of  its  intension.  But  it  will  not 
be  necessary  for  practical  purposes  to  consider  more  than  one 
of  them.  The  scheme  for  the  quantity  of  extension  is  the  one 
usually  employed,  and  as  it  avails  to  represent  and  test  all 
practical  cases  of  reasoning,  and  the  relations  between  terms, 
we  shall  confine  our  method  to  it  alone.  It  is  the  symboliza- 
tion  for  the  quantity  of  extension  that  is  used  to  explain  the 
distribution  of  terms,  which  is  the  next  topic  for  considera- 
tion. 

2d.  The  Distribution  of  Subject  and  Predicate. — By 
the  distribution  of  a  term  we  mean  that  something  is  said 
about  the  whole  of  what  it  contains  ;  that  is,  about  the  whole 
of  its  extension.  Thus  in  the  proposition  "  All  men  are  mor- 
tal "  we  state  something  about  the  whole  class  of  men,  and 
hence  the  subject  in  this  proposition  is  said  to  be  distributed. 
An  undistributed  term,  therefore,  is  one  in  which  we  do  not 
say  something  about  the  whole  class  denoted  by  the  term. 
Thus  in  proposition  I,  "  Some  men  are  negroes,"  we  do  not  as- 
sert something  of  the  whole  class  of  men,  and  hence  the  sub- 
ject is  said  to  be  undistributed.  This  appears  very  clearly  in 
the  symbolization  we  have  adopted.  For  instance,  in  Fig.  5 
the  circle  S  represents  that  something  is  said  about  the  whole 
of  the  class  it  denotes,  and  so  in  any  other  figure  s}rmbolizing 
a  universal  proposition,  while  the  undistributed  character  of 
the  subject  in  particular  propositions  is  equally  evident  when 
allowance  is  made  for  their  proper  implications.  But  no  am- 
biguity in  regard  to  this  matter  will  be  noticed  in  Figs.  14,  15, 
16,  and  17. 

In  regard  to  the  predicate,  it  may  not  be  so  easy  to  deter- 
mine its  degree  of  distribution  from  the  figures  without  care- 
ful explanation.  But  this,  perhaps,  may  make  it  clear.  In 
proposition  A,  "  All  metals  are  elements,"  we  perceive  without 
difficulty  that  something  is  said  about  the  whole  of  the  sub- 


RELATION  BETWEEN  SUBJECT  AND  PREDICATE    139 

ject,  and  that  it  is  therefore  distributed.  But  nothing  is  said 
or  implied  about  the  whole  of  the  predicate,  except  as  repre- 
sented in  Fig.  4,  where  we  assume  that  the  proposition  is  a 
definition.  But  as  this  can  never  be  assumed  formally,  and  as 
all  propositions  must  be  formally  treated  in  Formal  Logic,  Fig. 
5  and  Fig.  14  are  the  only  proper  symbols  of  the  quantitative 
relation  between  subject  and  predicate.  Nothing,  then,  is 
definitely  said  about  the  whole  of  the  predicate  in  proposition 
A,  because  other  substances  besides  "metals"  may  be  included 
in  "  elements."  If  we  said  anything  about  the  whole  of  the 
class  "  elements  "  in  the  proposition  "  All  metals  are  elements," 
we  could  reverse  the  order  of  subject  and  predicate  and  say, 
"All  elements  are  metals."  But  if  other  substances  besides 
metals  are  elements  this  latter  proposition  could  not  be  true, 
and  hence  as  long  as  we  can  possibly  say  that  other  things  be- 
sides metals  are  elements,  or  as  long  as  the  proposition  "All 
metals  are  elements,"  is  entirely  silent  about  the  extension  of 
the  term  "  elements,"  we  have  asserted  nothing  about  the  whole 
of  it,  as  indicated  by  the  larger  circle  P  in  Fig.  5,  and  hence 
the  predicate  is  not  distributed.  In  proposition  I,  "Some 
metals  are  elements,"  the  same  conclusion  is  apparent,  and  the 
predicate  is  undistributed. 

In  proposition  E,  "No  men  are  trees,"  for  instance,  not  only 
is  something  said  about  the  whole  of  the  subject,  but  some- 
thing is  also  said  about  the  whole  of  the  predicate.  It  is  defi- 
nitely excluded  in  its  whole  extension  from  the  subject,  as  Figs. 
10  and  15  indicate  :  That  is,  "men"  are  not  any  part  of  the 
class  "  trees,"  so  that  the  whole  of  the  class  "  trees  "  is  ex- 
cluded from  the  subject,  and  we  can  as  well  say  "  No  trees  are 
men,"  as  "No  men  are  trees."  Hence  in  the  negative  proposi- 
tion E  the  predicate  is  distributed,  since  something  is  said  or 
denied  about  the  whole  of  it.  In  the  negative  proposition  O 
the  same  conclusion  will  be  apparent,  if  we  merely  observe  that 
a  part  of  the  subject  is  definitely  excluded  from  the  whole  of 
the  predicate,  as  is  clear  in  Figs.  12  and  17.  The  predicate  of 
O,  "  Some  elements  are  not  metals  "  is  therefore  distributed. 
We  therefore  summarize  the  rules  for  the  distribution  of  sub- 


140  ELEMENTS  OF  LOGIC 

ject  and  predicate  as  follows.    We  give  two  forms  of  statement, 
and  the  student  may  adopt  the  most  convenient : 

Subject.  Predicate. 

f  TT   •          i    j  Affirmative  A.         Distributed.  Undistributed. 

J  universal   -  Negative      E          Distributed.  Distributed. 

Propositions                         j  Affirmative  I.          Undistributed.  Undistributed. 

[  particular  (  Negative       Q.         Undistributed.  Distributed. 

All  Universal  propositions,  A  and  E,  distribute  the  subject. 

All  Particular  propositions,  I  and  O,  do  not  distribute  the 
subject. 

All  Affirmative  propositions,  A  and  I,  do  not  distribute  the 
predicate. 

All  Negative  propositions,  E  and  O,  distribute  the  predicate* 

The  symbol  which  I  shall  adopt  to  indicate  the  distribu- 
tion of  a  term  will  be  a  small  circle  placed  around  the  sub- 
ject or  predicate,  as  the  case  may  be.  Thus  the  distribution 
and  non-distribution  of  terms  in  A,  E,  I,  and  O  may  be  repre- 
sented as  follows,  the  cross  indicating  a  negative  proposition  : 
A,®  =  P.     E,  ®X®.     I,  S  =  P.     O,  Sx®. 

*  General  references  on  the  relation  between  subject  and  predicate  are 
the  following  :  Venn :  Empirical  Logic,  Chaps.  VIII.  and.  IX.  ;  Symbolic 
Logic,  Chaps.  I.  and  VII.,  inclusive;  Bosanquet :  Logic,  Book  I.,  Chaps. 
I.  and  VII.,  inclusive;  Keynes:  Formal  Logic,  Part  II.,  Chap.  VI.; 
Wundt :  Logik,  Dritter  Abschnitt,  Cap.  I.  and  II. 


CHAPTER  IX. 

OPPOSITION 

1st.  Meaning  of  Opposition. — Opposition  treats  of  the 
relations  between  the  propositions  A,  E,  I,  and  O,  growing 
out  of  their  quantity  and  quality ;  that  is,  out  of  the  fact  that 
they  are  universal  and  particular  on  the  one  hand,  and  affirm- 
ative and  negative  on  the  other.  It  has  not  to  do  with  the 
subject  and  predicate,  or  the  elements  of  the  proposition  as 
such,  but  with  the  propositions  as  a  whole.  When  they  con- 
tain the  same  matter  they  have  certain  relations  of  agreement 
or  conflict  which  it  is  the  business  of  the  logician  to  exhibit. 
Unless  they  do  contain  the  same  matter  no  such  relation  can 
be  determined.  We  can,  in  such  cases,  only  decide  upon  their 
quantity  and  quality,  and  so  merely  treat  them  as  universal  or 
particular,  affii*mative  or  negative.  But  the  various  relations 
of  agreement  and  conflict  between  conceptions  give  rise  to 
corresponding  relations  between  propositions  containing  them, 
and  hence  we  require  to  ascertain  the  rules  which  regulate  the 
extent  to  which  any  given  proposition  is  true  or  false  when 
another  is  known  to  be  true  or  false.  Some  propositions,  if 
true,  interfere  with  the  truth  of  others,  and  some  do  not.  On 
the  other  hand,  some,  if  false,  necessitate  the  truth  of  others, 
and  some  do  not.  The  sense  and  extent  to  which  this  is  true 
remains  to  be  determined. 

If  "  All  horses  are  animals,"  it  cannot  be  true  at  the  same 
time  that  "  No  horses  are  animals,"  or  that  "  Some  horses  are 
not  animals."  This  we  express  by  saying  that  if  A  is  true,  E 
and  O  cannot  be  true  at  the  same  time  ;  it  is  inconsistent  with 
both  of  them.  Also,  if  it  be  true  that  "  No  men  are  quadru- 
peds," it  cannot  be  true  that  "All  men  are  quadrupeds,"  or 
that  "  Some  men  are  quadrupeds."     This  we  again  express  by 


142  ELEMENTS  OF  LOGIC 

saying  that  if  E  be  true,  A  and  I  cannot  be  true  at  the  same 
time ;  it  is  inconsistent  with  them.  But  it  is  important  to 
observe  that  if  "  All  men  are  quadrupeds  "  is  false,  it  follows 
that  "Some  men  are  not  quadrupeds."  It  may  be  true,  also, 
that  "No  men  are  quadrupeds,"  but  the  falsity  of  the  universal 
affirmative  with  the  same  terms  does  not  prove  that  fact.  It 
can  only  prove  the  truth  of  the  particular  negative,  and  it  re- 
mains entirely  unknown  from  that  proposition  whether  its 
universal  negative  is  true  or  false.  Hence  if  A  be  false,  it  fol- 
lows that  O  must  be  true,  but  it  does  not  follow  that  E  is  true 
or  false.  Now  if  it  be  false  that  "  Some  men  are  not  mortal," 
it  must  follow  that  "All  men  are  mortal,"  and,  as  we  have 
shown  in  the  first  case,  the  negative  of  this,  namely,  "  No  men 
are  mortal,"  is  false.  This  we  express  by  saying  that  if  O  be 
false,  A  is  true  and  E  is  false.  Similarly,  if  I  be  false,  E  must 
be  true  and  A  must  be  false.  In  this  way  we  find  that  if  A 
be  true,  O  is  false,  and  if  A  be  false,  O  is  true  ;  again,  if  E  be 
true,  I  is  false,  and  if  E  be  false,  I  is  true.  On  the  other  hand, 
if  O  be  true,  A  is  false,  and  if  O  be  false,  A  is  true  ;  and  if  I  be 
true,  E  is  false,  and  if  I  be  false,  E  is  true.  This  kind  of  incon- 
sistency between  A  and  O,  on  the  one  hand,  and  E  and  I,  on 
the  other,  we  call  contradiction.  In  a  loose  sense  the  words 
"  contradiction  "  and  "  contradictory  "  are  used  to  express  any 
kind  of  inconsistency  which  prevents  two  things  from  being 
true  at  the  same  time.  But  the  relations  between  propositions 
A  and  E  are  so  different  from  those  between  A  and  O,  and  E 
and  I,  that  the  term  "  contradictory  "  has  been  chosen  to  indi- 
cate that  mutual  inconsistency  between  A  and  O,  and  E  and 
I,  by  which  only  one  of  them  can  be  true,  and  only  one  of 
them  false,  at  the  same  time.  But  A  and  E  are  called  Con- 
traries, because  although  the  truth  of  A  implies  the  falsity  of 
E,  and,  vice  versa,  the  truth  of  E  implies  the  falsity  of  A,  yet 
the  falsity  of  A  does  not  imply  the  truth  of  E,  nor  the  falsity 
of  E  the  truth  of  A  The  mutual  inconsistency  existing  be- 
tween the  universals  and  their  opposite  particulars  does  not 
exist  between  universals.  Hence  they  are  called  oj^posites  to 
distinguish  them  from  contradictories. 


OPPOSITION  143 

It  remains  to  determine  the  relations  between  A  and  I,  E 
and  0,  and  I  and  0.  If  it  be  true  that  "  All  men  are  mortal," 
it  must  be  true  that  "  Some  men  are  mortal."  So,  if  it  be 
true  that  "  No  men  are  trees,"  it  must  be  true  that  "  Some 
men  are  not  trees."  This  we  express  by  saying  that  if  A  be 
true,  I  is  true,  and  if  E  be  true,  O  is  true,  because  the  part 
must  be  included  in  the  whole.  But  if  it  be  true  that  "  Some 
men  are  wise,"  it  does  not  follow  that  "All  men  are  wise  ;  " 
and  if  it  be  true  that  "  Some  men  are  not  wise,"  it  does  not 
follow  that  "  No  men  are  wise."  This  we  express  by  saying 
that  if  I  be  true,  A  is  indeterminate,  and  if  O  be  true,  E  is  in- 
determinate. This  is  because  we  can  affirm  nothing  of  the 
whole  when  we  affirm  something  only  of  the  part.  If  we  were 
to  take  cases  supposing  the  falsity  of  A,  we  should  find  I  in- 
determinate, or  the  falsity  of  E,  we  should  find  O  indetermin- 
ate. But  the  falsity  of  I  does  not  leave  A  indeterminate,  nor 
does  the  falsity  of  O  leave  E  indeterminate.  This  variable 
relation  is  expressed  by  calling  A  and  I,  or  E  and  O,  subalterns 
of  each  other.  But  I  and  O  are  called  subalternates,  and  A 
and  E  are  each  called  a  subalternans. 

When  we  compare  I  and  O  we  find  that  they  represent 
propositions  of  opposite  quality  ;  that  is,  one  is  affirmative 
and  the  other  negative,  and  in  that  respect  they  are  the  oppo- 
site of  each  other.  But  the  relation  between  them  is  the  re- 
verse of  that  between  A  and  E.  If  it  be  true  that  "  Some 
metals  are  elements,"  the  law  of  contradiction  already  estab- 
lished between  I  and  E  will  make  the  proposition  "  No  metals 
are  elements "  false,  and  by  subalternation  O,  "  Some  metals 
are  not  elements,"  will  be  indeterminate.  That  is,  nothing 
follows  about  O  from  the  truth  of  I,  and  also  nothing  about  I 
from  the  truth  of  0.  But  if  it  be  false  that  "  Some  men  are 
not  trees,"  it  follows  by  contradiction  that  the  proposition 
"  No  men  are  trees  "  is  true,  and  by  subalternation,  "  Some 
men  are  not  trees,"  is  true  also.  This  we  express  by  saying 
that  if  I  be  false,  O  is  true,  and  if  O  be  false,  I  is  true.  But 
both  cannot  be  false  at  the  same  time,  and  both  may  be  true. 
This  relation  is  expressed  by  calling  them  subcontraries,  as  A 


144  ELEMENTS  OF  LOGIC 

and  E  are  called  contraries  because  they  cannot  both  be  true 
at  the  same  time,  but  both  may  be  false. 

These  various  relations  of  the  propositions  A,  E,  I,  and  O 
are  represented  by  a  diagram  which  has  been  but  slightly 
modified  since  Aristotle.  It  is  called  the  Square  of  Opposi- 
tion. 

A       Contraries.       E 


V* 


0>  <fr    S*  <B 

?>      ^  %  m 

I    Subcontraries.     O 

The  relations,  as  we  have  developed  them,  can  easily  be  ap- 
plied to  this  scheme.  They  are  embodied  in  the  following 
rules,  which  it  is  important  to  keep  in  mind  : 

1.  Of  contradictory  propositions,  one  must  be  true  and  the 

other  false. 

2.  Of  contrary  propositions,  both  cannot  be  true  at  the  same 

time,  and  both  may  be  false. 

3.  Of  subcontrary  propositions,  one  only  can   be  false,  and 

both  may  be  true  at  the  same  time. 

4.  Of  subalterns,  both  may  be  true  and  both  may  be  false  at 

the  same  time.     But  if  the  subalternans  be  true,  the 
subalternate  is  true,  and  if  the  subalternate  be  false 
the  subalternans  is  false  ;  if  the  subalternans  be  false, 
the  subalternate  is  indeterminate,  and  if  the  subalter- 
nate be  true,  the  subalternans  is  indeterminate. 
2d.  Application    of   Opposition    and    its    Principles. — 
Nothing  can  be  determined  formally  about  the  relation  of  op- 
position between  the  propositions  A,  E,  I,  and  O,  unless  we 
assume  identity  of  matter.     A  difference  of  matter  simply  iso- 
lates the  two  propositions  and  throws  them  out  of  all  relation 
to  each  other  in  the  scheme  of  opposition.     The  question  is 
therefore  suggested,  What  laws  determine  the  nature  of  these 
relations  of  contradiction,  contrariety,  and  subalternation,  and 
what  must  be  taken  into  account  when  applying  them  to  actual 


OPPOSITION  145 

discourse  ?  Keynes  furnishes  a  complete  answer  to  the  first 
of  these  questions. 

"  The  inferences,"  he  says,  "  based  on  the  square  of  opposition, 
may  be  considered  to  depend  exclusively  on  the  three  funda- 
mental Laws  of  Thought,  namely,  the  Law  of  Identity — A  is 
A  ;  the  Law  of  Contradiction — A  is  not  A ;  and  the  Law  of  Ex- 
cluded Middle — A  is  either  B  or  not  B."  For  example,  from 
the  truth  that  "All  men  are  mortal"  I  may  infer  by  the  Law  of 
Identity  that  "  Some  men  are  mortal,"  and  by  the  Law  of  Con- 
tradiction the  falsity  of  the  proposition  that  "  Some  men  are 
not  mortal."  By  the  Law  of  Excluded  Middle  we  can  infer 
from  the  falsity  of  the  proposition  "  All  men  are  mortal,"  the 
truth  of  the  proposition  that,  "Some  men  are  not  mortal." 
The  Law  of  Identity  means  that  a  thing-  can  be  affirmed  of  it- 
self, or  conceptions  which  agree  with  each  other  can  be  af- 
firmed in  that  sense.  The  Law  of  Contradiction  means  that 
two  conflicting  or  contradictory  conceptions  cannot  be  af- 
firmed of  a  thing  at  the  same  time.  Thus  I  cannot  affirm  that 
a  man  is  both  "  mortal "  and  "  not  mortal  "  at  the  same  time. 
The  Law  of  Excluded  Middle  means  that  of  two  contradictories 
one  must  be  true.  But  a  fuller  discussion  of  these  laws  must 
be  postponed  to  a  later  chapter.  This  brief  account  will  suf- 
fice for  the  applications  with  which  we  have  to  deal  at  pres- 
ent. 

In  discourse  and  controversy  we  have  to  be  careful  about 
the  real  nature  of  our  conceptions  and  propositions.  We  are 
liable  to  mistake,  at  times,  a  contrary  for  a  contradictory  judg- 
ment, or  an  indefinite  for  a  definite  judgment,  or  subcontra- 
ries  for  contraries.  This  will  particularly  be  the  case  when 
propositions  are  one  thing  in  form  and  another  thing  in  mat- 
ter. Thus  singular  propositions  are  treated  as  universal  in 
form  and  are  therefore  contraries.  But  in  matter,  subject  and 
predicate  being  the  same  and  their  quality  the  opposite  of  each 
other,  they  are  contradictories;  as,  for  example,  "Socrates  is  a 
man,"  the  only  possible  negative  of  which  is  "Socrates  is  not  a 
man."  Pure  universals  have  two  opposites,  the  contrary  and  the 
contradictory,  but  singulars  have  only  one,  which  is  in  reality 
10 


146  ELEMENTS  OF  LOGIC 

the  contradictory,  as  will  be  seen  in  the  case  given.  Thus  if  it 
be  true  that  "  Socrates  is  a  man,"  the  negative,  "Socrates  is  not 
a  man,"  is  false,  and  vice  versa.  So  far  it  seems  like  a  case  of 
contraries.  But  if  "  Socrates  is  a  man "  be  false,  it  is  true 
that  "  Socrates  is  not  a  man,"  and  vice  versa.  This  makes  it  a 
case  of  contradiction,  because  if  they  were  contraries  they 
might  both  be  false  at  the  same  time.  This  would  mean  that 
the  assumed  or  proved  falsity  of  the  proposition  "  Socrates  is 
a  man,"  would  leave  the  proposition  "  Socrates  is  not  a  man  " 
indeterminate.  But  we  observe  that  it  cannot  be  so,  and  hence 
singular  judgments  in  respect  to  quality  have  to  be  treated  as 
contradictories,  but  in  respect  to  quantity  as  universals.  This 
will  determine  the  relation  between  the  two  propositions 
"  Socrates  is  a  man  "  and  "  Socrates  is  a  horse."  They  must 
be  regarded  as  contraries,  not  as  contradictories.  They  con- 
tain different  matter  in  their  predicates,  but  the  same  matter 
in  their  subjects  ;  so  that  although  the  predicates  are  both 
positive  concepts  they  are  mutually  exclusive  as  species,  and 
so  relatively  negative  in  conrparison  with  each  other.  The 
two  propositions  are  related  as  contraries  because  the  truth 
of  either  denies  the  other,  while  both  may  be  false.  If  it 
be  true  that  "  Socrates  is  a  man,"  it  cannot  be  true  that 
"  Socrates  is  a  horse,"  and  so  if  he  were  a  horse  he  could  not 
be  a  man.  But  if  it  be  false  that  "  Socrates  is  a  man,"  it 
does  not  follow  that  he  is  a  horse,  because  he  might  be  any- 
thing else  except  a  man.  Hence  terms  representing  co-ordi- 
nate species  will  be  contraries,  not  contradictories,  in  the 
scheme  of  opposition. 

But  what  will  be  made  of  the  propositions  "  Socrates  is  a 
man  "  and  "  Socrates  is  a  Greek  ?  "  Of  course,  formally  neither 
these  nor  the  previous  propositions  can  be  treated  of  under 
the  principles  of  opposition,  and  I  am  not  designing  so  to 
treat  them.  I  am  endeavoring  to  give  the  purely  formal  rules 
some  modifications  to  suit  their  material  application.  The  last 
two  propositions,  although  singular,  as  the  two  previous  ones, 
are  somewhat  different  because  of  the  relation  between  the 
two  predicates.     These,  instead  of  being  co-ordinate  species, 


OPPOSITION  147 

are  genus  and  species.  Hence  if  the  first  be  true  it  does  not 
follow  that  the  second  is  either  true  or  false  ;  but  if  it  be  false 
the  second  is  false.  It  is  interesting  to  note  from  this  that 
they  are  subalterns,  the  proposition  "Socrates  is  a  man"  being 
the  subalternate,  and  "  Socrates  is  a  Greek  "  being  the  subal- 
ternans.  We  may  generalize,  therefore,  in  such  cases,  by  say- 
ing that  in  singular  judgments  with  the  same  subject,  when 
the  predicate  is  a  genus  in  one  and  a  species  in  the  other,  the 
genus  is  the  mark  of  the  subalternate  and  the  species  of  the 
subalternans. 

When  the  predicate  is  the  same  the  subjects  can  never  be  a 
genus  and  species,  and  the  propositions  remain  singular  at  the 
same  time.  Hence  no  relation  of  opposition  in  such  cases  is 
determinable.  Thus,  in  the  propositions,  "Socrates  is  a 
Greek,"  and  "Plato  is  a  Greek,"  nothing  can  be  said  about 
the  truth  or  falsity  of  one  when  the  other  is  either  true  or 
false. 

Hence,  when  the  predicates  are  the  same  in  both  propo- 
sitions, and  the  subject  of  one  a  genus,  and  of  the  other  a  spe- 
cies, both  propositions  formally  are  universals,  but  materially 
they  are  A  and  I,  or  universal  and  particular,  and  so  are  sub- 
alterns again.  But  in  this  case  the  genus  marks  the  subalter- 
nans and  the  species  the  subalternate.  Thus  "  All  Greeks  are 
men,"  and  "  Socrates  is  a  man."  The  truth  of  the  first  implies 
that  of  the  second,  and  the  falsity  of  the  first  leaves  the  second 
indeterminate  (except  on  a  condition  to  be  discussed  again), 
while  the  truth  of  the  second  leaves  the  first  indeterminate, 
and  the  falsity  of  the  second  implies  the  falsity  of  the  first. 
We  must  keep  in  mind,  however,  that  we  assume  all  along 
that  Socrates  is  a  Greek,  and  hence  an  individual  or  a  species 
of  the  genus. 

The  matter  is  still  more  complicated  when  we  come  to  con- 
sider universal  propositions  or  judgments.  Suppose  we  take 
the  first  example  with  the  same  subject  and  with  co-ordinate 
species  for  predicates.  Thus  "  All  men  are  bipeds,"  and  "All 
men  are  rational."  It  is  apparent  in  such  cases  that  neither 
agreement  nor  conflict  between  them  can  be  inferred  from 


148  ELEMENTS  OF  LOGIC 

either  the  truth  or  the  falsity  of  one  of  them.  Hence  a  rela- 
tion of  opposition  is  not  determinate  here,  any  more  than  in 
the  singular  propositions  "  Socrates  is  a  man,"  and  "  Socrates 
is  blind."  If  the  predicates  be  such  inconsistent  concepts  as 
" quadrupeds "  and  "newspapers,"  the  propositions  might  be 
treated  as  contraries,  but  there  is  no  criterion  for  determining 
this  opposition.  It  requires  to  be  a  uniform  concomitant  of 
some  other  characteristic.  But  if  the  predicates  be  genus  and 
species,  and  the  subject  identical,  as  "All  men  are  vertebrates," 
and  "  All  men  are  bipeds,"  the  problem  is  not  different  from 
the  previous  one,  because  genus  and  species,  or  conferentia 
and  differentia,  always  agree  and  never  conflict,  but  never 
imply  each  other.  No  relation  of  opposition  therefore  can  be 
established  between  such  propositions. 

If  the  predicate  remain  the  same,  and  the  subjects  are  genus 
and  species,  as  "All  Europeans  are  Caucasians,"  and  "All 
Frenchmen  are  Caucasians,"  the  propositions  are  evidently 
snbalterns,  the  genus  marking  the  subalternans,  and  the  spe- 
cies the  subalternate,  as  already  shown.  But  as  between  "  All 
Europeans  are  Caucasians,"  and  "  Some  Frenchmen  are  not 
Caucasians,"  we  evidently  have  contradictories,  and  so  also 
with  "  All  Frenchmen  are  not  Caucasians,"  because  the  relation 
between  genus  and  species  determines  the  particularity  of  the 
last  proposition  in  comparison  with  the  first.  Of  course  such 
propositions  as  "All  men  are  bipeds,"  and  "All  men  are  quad- 
rupeds," or  "  All  wise  men  are  good,"  and  "All  wise  men  are 
bad,"  will  be  contraries,  owing  to  the  nature  of  the  concepts 
biped  and  quadruped,  and  good  and  bad.  But  the  form  of  the 
terms  cannot  determine  this  fact,  and  hence  no  rule  for  esti- 
mating them  can  be  established.  It  will  be  important  to  ob- 
serve, however,  that  good  and  bad,  wise  and  ignorant,  rational 
and  irrational,  beautiful  and  ugly,  etc.,  are  often  used  as  con- 
tradictory conceptions,  and  sometimes  only  as  contraries.  The 
latter  is  the  true  conception  of  them,  as  there  is  a  third  alter- 
native between  them,  owing  to  the  fact  that  they  are  both 
positive  terms.  A  true  contradiction  can  exist  only  between 
positive  and  negative  terms,  never  between  two  positives,  or 


OPPOSITION  149 

between  a  positive  and  a  nego-positive.  The  latter  are  only 
contraries,  and  so  establish  contrariety  between  propositions 
having  them  as  predicates,  but  with  the  same  subject. 

It  is  very  important  to  keep  these  facts  and  distinctions  in 
mind  in  order  to  understand  the  mental  processes  involved  in 
one  man's  asserting  that  "  Washington  was  a  good  man,"  and 
another  controverting  it  by  the  counter-assertion  that  "  Wash- 
ington stole  a  horse,"  or  one  man's  affirming  that  "  Americans 
are  a  mercantile  people,"  and  another's  asserting  that  "  A.  B. 
C.  of  the  Americans  are  great  scientists."  In  the  first  two  the 
propositions  are  evidently  contraries.  In  the  second  instances, 
everything  depends  upon  whether  the  indefinite  proposition, 
"Americans  are  a  mercantile  people,"  be  regarded  as  a  univer- 
sal or  a  particular,  A  or  I.  If  the  former,  they  are  contraries, 
supposing  that  commerce  and  science  are  incompatible  call- 
ings, as  in  practice,  generally  at  least,  they  seem  to  be,  al- 
though perhaps  not  ideally  or  theoretically  so.  But  if  the 
proposition  be  I,  they  are  subcontraries,  under  the  same  as- 
sumption as  before.  But  the  cases  only  show  how  we  are  to 
analyze  the  conceptions  when  a  controversy  is  involved. 

In  the  relation  between  subcontrary  judgments  we  must  be 
careful  not  to  admit  the  ambiguous  use  of  the  word  some,  as 
this  would  at  once  imply  the  truth  of  the  opposite  proposition. 
But  in  the  true  conception  of  opposition  I  never  implies  O, 
and  O  never  implies  I.  Hence  the  term  "  some  "  must  always 
mean  a  part,  and  it  may  or  may  not  be  all,  as  has  ah-eady  been 
shown. 

Errors  in  argument,  so  far  as  an  assumption  of  the  relation 
between  propositions  may  be  concerned,  are  occasioned  in  two 
ways  ;  first,  in  assuming  an  agreement  between  them  when  they 
are  inconsistent,  and  second,  in  assuming  a  conflict  when  they 
are  consistent  with  each  other.  This  is  ordinarily  called  the 
fallacy  of  f</>i<>ra/t<>  Elenchi,  which  we  shall  have  to  explain 
again.  But  it  will  be  illustrated  in  the  course  of  the  present 
discussion.  We  are  to  examine  proof  and  disproof  of  judg- 
ments so  far  as  that  can  be  accomplished  under  the  scheme  of 
opposition. 


150  ELEMENTS  OF  LOGIC 

By  the  principles  of  opposition  we  can  only  prove  I  and  O  as 
subalternates  by  assuming  or  proving  A  and  E,  and  disprove 
A  and  E  by  assuming  or  proving  I  and  O  as  their  contradic- 
tories. Thus  we  prove  I  if  we  admit  A,  and  we  prove  O  if  we 
admit  E.  But  we  disprove  A  if  we  admit  O,  and  disprove  E 
if  we  admit  I.  In  such  formal  propositions  as  "  Some  men  are 
mortal "  and  "  All  men  are  mortal,"  this  is  evident  from  the 
rules.  But  in  such  proj^ositions  as  "  Europeans  are  white," 
and  "  Germans  are  white,"  it  is  not  evident  until  we  notice 
the  logical  relation  existing  between  the  conceptions  "  Euro- 
peans "  and  "  Germans."  I  prove  the  truth  of  the  latter  when 
I  admit  the  former,  assuming  that  "  Germans  "  are  a  species  of 
the  genus  "  Europeans,"  because  they  are  related  as  subalterns, 
the  former  being  the  subalternans,  and  the  latter  the  subalter- 
nate,  the  proposition  to  be  proved.  I  may  prove  that  "  Scien- 
tists are  learned,"  by  assuming  or  proving  that  "  Educated  men 
are  wise,"  if  "  learned  "  and  "  wise  "  are  identical,  and  "  educated 
men  "  be  the  genus  of  which  "  scientists  "  are  the  species.  But 
I  cannot  prove  that  "  Germans  are  good "  by  assuming  or 
proving  that  "  Europeans  are  not  bad,"  without  making  the 
second  assumjDtion  that  "  good"  and  "  not  bad  "  are  identical. 
In  disproof  the  matter  is  a  more  important  and  interesting 
one.  Here  we  have  to  do  with  A  and  E  as  conclusions,  and  I 
and  O  as  conditions  of  their  validity  or  invalidity.  "  In  order 
to  prove  the  falsity  of  A,  it  is  sufficient  to  establish  the  truth 
of  O,  and  it  is  superfluous,  even  if  possible,  to  prove  E  ;  simi- 
larly E  is  disproved  by  proving  I,  and  it  is  suj)erfluous  to 
prove  A.  Any  person  who  asserts  a  universal  proposition, 
either  A  or  E,  lays  himself  under  the  necessity  of  explaining 
away  or  disproving  every  single  exception  brought  against  it. 
An  opponent  may  always  restrict  himself  to  the  much  easier 
task  of  finding  instances  which  apparently  or  truly  contradict 
the  universality  of  the  statement,  but  if  he  takes  upon  himself 
to  affirm  the  direct  contrary,  he  is  himself  open  to  easy  at- 
tack. "Were  it  to  be  asserted,  for  instance,  that  'All  Chris- 
tians are  more  moral  than  Pagans,'  it  would  be  easy  to  adduce 
some  examples  showing  that  '  Some  Christians  are  not  more 


OPPOSITION  151 

moral  than  Pagans,'  but  it  would  be  absurd  to  suppose  that  it 
would  be  necessary  to  go  to  the  contrary  extreme,  and  show 
that  '  No  Christians  are  more  moral  than  Pagans.'  In  short, 
A  is  sufficiently  and  best  disproved  by  O,  and  E  by  I.  It  will 
be  easily  apparent  that,  vice  versa,  O  is  disproved  by  A,  and  I 
by  E  ;  nor  is  there,  indeed,  any  other  mode  of  disproving 
these  particular  propositions."  The  error  in  disproof,  how- 
ever, may  He  in  certain  assumptions  about  the  relations  be- 
tween the  two  propositions  after  the  proper  one  has  been 
proved.  Thus  it  may  be  no  disproof  of  the  assertion  that 
"  John  Smith  is  good,"  to  prove  that  he  is  not  civilized,  be- 
cause the  conceptions  "  good  "  and  "  not  civilized  "  are  not 
necessarily  contradictory,  nor  even  contrary.  I  simply  evade 
the  issue.  Again,  it  is  no  disproof  of  the  assertion  that 
"  Cromwell  was  a  usurper,"  to  say  that  "  Foreign  nations  ac- 
knowledged his  authority,"  any  more  than  it  would  be  a  proof 
of  his  legitimacy  to  make  the  same  statement.  Likewise  it  is 
no  disproof  of  a  man's  badness  to  say  that  he  is  religious,  any 
more  than  we  should  prove  he  is  white  by  showing  that  he  is 
not  black.  If  I  assert  that  "  Government  is  a  necessary  insti- 
tution," it  is  no  disproof  of  it  to  show  that  some  governments 
are  bad.  Many  arguments,  however,  are  conducted  upon  just 
such  logic,  where  agreement  or  conflict  are  simply  assumed 
by  both  parties  to  exist  between  conceptions  which  may  be 
inconsistent  in  certain  accidental  relations,  but  not  necessarily 
so.  Thus,  if  I  assert  that  free-trade  is  the  right  policy  for  a 
country,  it  is  no  disproof  of  it  to  show  that  protection  helps 
manufacturers  ;  nor  is  it  a  disproof  of  it  to  show  that  I  have 
myself  instituted  a  policy  of  protection.  The  untrained  mind, 
however,  is  likely  to  suppose  the  argument  valid  in  both 
cases.  In  purely  formal  Logic  this  is  all  very  easy,  but  in 
applied  Logic,  or  actual  discourse,  we  require  dexterity  in  dis- 
covering the  relations  between  conceptions  in  order  to  test 
the  connection  of  general  principles  with  them.  To  illus- 
trate, take  the  conceptions  just  mentioned  in  the  last  exam- 
ple, protection  and  free-trade.  Without  some  knowledge  of 
economics,  perhaps,  we  should  not  suspect  or  detect  any  rola- 


152  ELEMENTS  OF  LOGIC 

tion  of  either  agreement  or  conflict.  "Protection,"  as  a 
formal  and  logical  conception,  might  suggest  nothing  more 
than  the  idea  of  "self-defence"  or  "self-preservation,"  and  so 
carry  everything  before  it  by  this  association,  while  no  con- 
ceivable meaning  might  be  attached  to  "  free-trade,"  except 
the  vague  sense  of  contradiction  or  opposition  from  the  mere 
fact  that  some  one  so  considered  it.  But  an  examination  of 
their  real  meaning  from  a  knowledge  of  economics  shows  them 
to  be  contradictories  in  reality,  and  not  merely  contraries,  al- 
though they  are  both  positive  terms,  in  perhaps  every  other 
connection  than  this  one.  This  is  made  clear  by  the  fact  that 
"  protection "  is  a  restriction  upon  free  commerce,  "  free- 
trade  "  is  the  absence  of  that  restriction.  We  see  in  this  dis- 
tinction the  presence  and  the  absence  of  certain  qualities 
which  defines  respectively  a  positive  and  negative  term. 

In  many  cases  the  error  can  be  tested  by  taking  into  ac- 
count the  assumptions  made  in  any  particular  assertion.  But 
this  introduces  the  syllogism  into  the  problem,  while  it  may 
be  possible  to  decide  the  matter  upon  the  principles  of  oppo- 
sition. This  is  all  that  we  have  here  considered,  and  the  stu- 
dent should  practise  the  application  of  these  principles  on 
every  possible  occasion,  not  so  much  for  the  sake  of  familiar- 
ity with  the  laws  of  formal  Logic,  as  for  a  better  understand- 
ing of  the  means  for  dealing  with  the  subject-matter  of  dis- 
course and  controversy.  As  an  illustration  of  what  a  mental 
process  is,  and  what  law  of  opposition  is  involved  or  implied, 
I  quote  at  random  a  statement  from  an  article  in  a  monthly 
periodical :  "  Those  who  oppose  nationalism  on  the  ground 
that  the  present  social  condition  is,  by  reason  of  its  privations, 
a  blessing,  ought  to  understand  that  the  exact  opposite  of  a 
false  proposition  is  by  no  means  certain  to  be  the  true  one — 
though  it  is  a  favorite  argumentative  short-cut  to  assume  this 
to  be  the  case."  Now  whether  he  knew  it  or  not,  the  writer 
was  using  the  law  regulating  the  relation  between  contraries 
in  asserting  that  "the  exact  opposite  of  a  false  proposition 
was  by  no  means  certain  to  be  the  true  one."  But  the  asser- 
tion charges  a  fallacy  upon  his  opponent  which  would  have 


OPPOSITION  153 

been  technically  called  an  ignoratio  elenchi.  The  case  could, 
perhaps,  have  been  made  stronger  by  connecting  the  error  di- 
rectly with  the  violation  of  a  law  of  Logic.  Illustrations  of 
the  same  kind  are  plentiful,  and  the  student  can  profitably 
spend  his  time  in  looking  for  them.* 

*  Special  references  on  the  subject  of  Opposition  need  not  be  indicated, 
as  the  general  agreement  upon  it  leaves  no  points  of  importance  in 
doubt.  The  works  previously  mentioned  deal  with  the  subject  very 
briefly. 


CHAPTER  X. 

IMMEDIATE   INFERENCE 

1st.  Definition  and  Divisions. — An  inference  of  any  kind 
is  merely  the  explicit  statement  of  what  is  implicit  in  a  pre- 
vious assertion  or  thought.  If  this  account  of  it  is  not  clear, 
we  may  define  it  as  drawing  one  conception  or  truth  from  an- 
other, which  implies  the  one  drawn,  or  it  is  the  carrying  out 
into  a  following  proposition  a  thought  which  was  virtually 
contained  in  an  antecedent  judgment.  This  last  definition  is 
Hamilton's,  slightly  modified.  It  can  be  illustrated  in  several 
ways.  I  can  infer  that  "  Europeans  are  white,"  if  I  know  that 
Caucasians  are  white,  and  that  Europeans  are  Caucasians.  Or 
I  may  infer  that  the  weather  will  be  clear,  when  I  see  the 
clouds  breaking.  The  first  form  of  inference  is  said  to  be  de- 
ductive, and  the  second  inductive.  But  for  the  present  we 
need  only  to  know  that  a  certain  fact  or  facts  may  suggest  to 
consciousness  other  facts  not  explicitly  stated  or  conceived  in 
the  first  case,  and  which  are  said  to  follow  from  it.  An  infer- 
ence, then,  is  what  follows  from  another  thing,  in  so  far  as  the 
latter  is  a  conceived  truth.  A  case  quite  different  from  the  pre- 
vious illustrations  is  the  succeeding  one.  From  the  proposition 
"  The  sciences  are  useful  studies,"  I  can  infer  that  "  Some  of 
the  useful  studies  are  sciences,"  or  from  "  All  negroes  are 
black,"  that  "  Those  who  are  not  black  are  not  negroes."  It  is 
evident  that  these  forms  of  inference  are  quite  different  from 
the  previous  instances.  Hence  we  proceed  to  the  two  kinds 
of  inference. 

There  is,  first,  the  inference  known  as  Mediate  Inference,  or 
Reasoning,  which  is  illustrated  by  the  first  instances.     It  de- 


IMMEDIATE  INFERENCE  155 

notes  reasoning  by  the  agency  of  a  middle  term.  Thus  in  all 
mediate  inference  we  have  first  to  compare  two  terms  with  a 
third,  called  the  middle  term,  and  then  infer  that  these  two 
terms  can  be  compared  with  each  other.  But  it  is  quite  dif- 
ferent with  the  second  class  of  inferences.  This  is  called  Im- 
mediate Inference,  or  Reasoning,  and  denotes  reasoning  with- 
out the  use  of  a  middle  term.  Mediate  inference  requires  two, 
immediate  inference  but  one  proposition  as  a  basis.  This  is 
exemplified  in  the  illustrations.  Immediate  inference  is  usu- 
ally divided  into  Conversion,  Obversion,  Contraposition,  Added 
Determinants,  and  Complex  Conceptions.  But  I  shall  slightly 
modify  this  division  so  as  to  read  Conversion,  Obversion,  Con- 
traversion,  Inversion,  and  Contribution,  of  which  last  the  last 
two  in  the  previous  classification  are  modified  forms.  Oppo- 
sition is  not  treated  as  a  mode  of  inference.  But  it  would  not 
be  far  amiss  to  so  treat  it,  or  at  least  to  consider  it  as  involv- 
ing processes  of  immediate  inference.  For  we  pass  directly 
to  certain  conclusions  from  certain  other  conditions  or  pre- 
mises. But  it  differs  from  other  modes  of  inference  in  that  it 
requires  the  supposition  of  certain  material  truths  in  order  to 
render  it  possible,  while  in  other  forms  of  immediate  inference 
we  require  only  a  formal  relation  between  subject  and  predi- 
cate. Perhaps,  however,  if  we  regard  the  conditions  of  oppo- 
sition as  formal,  it  may  be  possible  to  give  the  transition  from 
A  to  0,  or  from  E  to  I,  by  contradiction,  or  from  A  to  E,  by 
contraries,  as  sufficiently  formal  and  inferential  to  speak  of 
the  process  as  involving  a  kind  of  immediate  inference.  Al- 
though I  believe  this  to  be  the  case  I  have  remained  by  tradi- 
tional usage  and  discussed  it  by  itself.  Under  immediate  in- 
ference, therefore,  I  begin  with  Conversion. 

2d.  Conversion. — Conversion  is  the  transposition  of  sub- 
ject and  predicate,  or  the  process  of  immediate  inference  by 
which  we  can  infer  from  a  given  proposition  another  having 
the  predicate  of  the  original  for  its  subject,  and  the  subject  of 
the  original  for  its  predicate.  But  there  are  certain  limita- 
tions under  which  the  transposition  can  take  place.  For  in- 
stance, from  the  proposition  that  "  All  horses  are  animals,"  I 


156  ELEMENTS  OF  LOGIC 

cannot  infer  that  "All  animals  are  horses; "nor  from  "All 
metals  are  elements "  that  "  Some  elements  are  not  metals," 
although  this  may  actually  be  the  case.  The  rules,  therefore, 
which  limit  the  process  of  conversion  are  two. 

(a)  The  quality  of  the  converse  must  be  the  same  as  that 

of  the  convertend. 

(b)  No  term  must  be  distributed  in  the  converse  which  is 

not  distributed  in  the  convertend. 

These  rules  may  be  abbreviated  so  as  to  read  :  Do  not 
change  the  quality  of  the  proposition,  and  Do  not  distribute  an 
undistributed  term.  We  may  indistribute  a  term  which  is  dis- 
tributed, but  not  vice  versa.  The  Convertend  is  the  proposi- 
tion to  be  converted :  the  Converse  is  the  converted  propo- 
sition. 

The  forms  of  Conversion  are  two,  according  as  the  quantity 
of  the  proposition  is  changed  or  remains  the  same.  If  the 
quantity  of  the  converse  remains  the  same  as  that  of  the  con- 
vertend, the  conversion  is  caUed  Conversio  simplex,  or  Simple 
Conversion  ;  if  the  quantity  is  changed,  it  is  called  Conversio 
per  accidens,  or  Limited  Conversion,  usually  Conversion  by 
Limitation.  We  have  only  to  illustrate  them  and  to  ascertain 
the  extent  of  their  application  to  the  several  propositions  A, 
E,  I,  and  O. 

Take  a  proposition  in  A,  "  All  apples  are  fruit."  In  this 
proposition,  as  already  shown,  the  predicate  is  not  distributed. 
This  means,  as  illustrated  in  Fig.  5  (p.  130),  that  other  things 
also  may  be  contained  in  the  class  "  fruit,"  so  far  as  can  be  de- 
termined by  the  assertion  given.  Hence,  if  in  transposing 
subject  and  predicate  we  say,  "  All  fruits  are  apples,"  we  shall 
be  asserting  more  than  the  original  proposition  will  permit. 
In  the  original  we  have  said  nothing  about  the  whole  of  the 
term  "  fruit,"  and  so  cannot  be  permitted  to  do  so  in  the  con- 
verse. Hence  we  can  assert  something  only  of  a  part  of  it, 
if  we  can  assert  anything  at  all.  That  we  may  assert  some- 
thing is  evident  from  the  fact  that  some  degree  of  identity  or 
connection  exists  between  the  subject  and  predicate  in  the 
convertend,   and   this   same  relation  can  be  asserted  in  the 


1 M MEDIA  TE  INFERENCE  157 

converse.  By  limiting  our  statement,  therefore,  to  the  part  of 
the  predicate  of  which  we  actually  affirm  something,  we  are  able 
to  infer  from  the  original  proposition  that  "  Some  fruits  are 
apples."  This  is  evidently  true,  if  the  original  be  true.  Here 
the  quantity  of  the  proposition  is  changed,  but  its  quality  re- 
mains the  same  ;  that  is,  the  quantity  of  the  convertend  is 
universal  and  its  quality  affirmative,  while  the  quantity  of  the 
converse  is  particular  and  its  quality  affirmative.  We  have, 
therefore,  converted  A  into  I.  To  convert  "  All  apples  are 
fruit"  into  "All  fruits  are  apples"  would  be  to  violate  the  sec- 
ond rule.  Hence  A  cannot  be  converted  into  A.  To  change 
the  quality  of  the  proposition  A  in  conversion,  that  is,  into 
either  E  or  O,  would  be  to  violate  the  first  rule,  and  it  is  ap- 
parent that  we  cannot  infer  an  exclusion  between  a  subject 
and  predicate  from  an  affirmed  connection  or  identity  between 
them.  Hence  A  cannot  be  converted  into  either  E  or  O,  and 
we  have  found  that  A  cannot  be  converted  into  A,  but  that  it 
can  be  converted  into  I,  which  is  to  say  that  propositions  in 
A  cannot  be  converted  simply,  but  only  by  limitation. 

There  is  one  exception  in  the  conversion  of  A.  This  is  the 
case  of  definitions,  and  of  singular  propositions  with  a  singu- 
lar term  for  predicate.  Definitions  are  universal  affirmatives, 
and  are  nevertheless  convertible  by  simple  conversion.  This 
is  a  case  of  A  being  converted  into  A.  But  this  is  only  be- 
cause the  nature  of  the  case  enables  us  to  know  the  extension 
or  distribution  of  the  predicate.  Formally  the  proposition  is 
like  all  others  and  not  convertible  simply.  But  materially  we 
know  from  the  fact  that  the  proposition  is  a  definition,  that 
the  extension  of  the  subject  and  predicate  is  the  same,  and 
hence  convertible  from  A  into  A.  Without  assuming  its  ma- 
terial nature,  however,  we  could  know  nothing  of  this  distri- 
bution, and  hence  formally  it  would  have  to  be  treated  as  all 
other  propositions  in  A,  which,  being  neither  formally  nor 
materially  distributed  in  the  predicate,  can  only  be  converted 
by  limitation.  Nevertheless  it  is  important  to  observe  that  in 
some  of  our  material  reasoning  the  mind  may  be  correct  in  its 
processes  on  the  ground  that  its  datum  is  a  definition  ;  that 


158  ELEMENTS  OF  LOGIC 

is,  subject  and  predicate  are  conceived  as  identical,  while  for- 
mally the  reasoning  may  seem  to  be  fallacious. 

The  exclusive  proposition,  although  it  appears  to  be  a  uni- 
versal, may  or  may  not  be  so.  The  "  only  "  means  some,  and 
it  may  and  it  may  not  be  all,  but  certainly  nothing  else.  When 
converted,  it  becomes  a  universal,  but  not  before  this. 

Proposition  I,  "  Some  men  are  vertebrates,"  can  only  be  con- 
verted simply,  or  by  Simple  Conversion.  We  cannot  infer 
from  it  that  "  All  vertebrates  are  men,"  for  the  same  reason 
that  we  could  not  convert  A  into  A.  It  is  because  the  predi- 
cate is  undistributed  in  the  convertend,  and  must  not  be  dis- 
tributed in  the  converse.  It  would  violate  rule  second  to  con- 
vert it  into  E  and  O,  and  for  the  same  reason  that  A  could  not 
be  converted  into  E  or  O.  Hence  I  must  be  converted  into  I, 
"  Some  men  are  vertebrates "  into  "  Some  vertebrates  are 
men."  This  is  Conversio  simplex,  or  Simple  Conversion,  be- 
cause the  form  of  the  converse  is  the  same  as  that  of  the  con- 
vertend. 

Proposition  E,  "  No  books  are  pens,"  can  be  converted  either 
simply  or  by  limitation.  In  this  the  predicate  is  distributed, 
and  this  fact  will  permit  of  its  distribution  in  the  converse. 
Besides,  since  something  is  said  excluding  the  whole  of  the 
predicate  from  the  subject,  we  can  assert  this  in  the  converse. 
Hence  we  can  infer  that  "  No  pens  are  books."  By  subalter- 
nation,  from  this  we  can  infer  "  Some  pens  are  not  books." 
The  first  is  the  simple,  and  the  second  the  limited  converse  of 
the  original.  Hence  E  is  convertible  into  either  E  or  O.  O 
might  be  called  a  weakened  converse  in  this  case,  because  E 
might  as  well  be  inferred. 

In  regard  to  propositions  in  O,  as,  "Some  men  are  not  Cau- 
casians," a  peculiar  difficulty  exists.  Eirst,  the  converse  must 
be  of  the  same  quality  as  the  convertend,  according  to  the  first 
rule.  It  must,  therefore,  be  negative.  But  second,  negative 
propositions  distribute  the  predicate.  Hence,  whether  we 
convert  "Some  men  are  not  Caucasians,"  by  simple  conversion 
or  by  limitation,  the  predicate  in  the  converse,  which  is  un- 
distributed in  the  convertend,  will  be  distributed,  and  hence 


IMMEDIATE  INFERENCE  159 

violates  the  second  rule.  0  cannot,  therefore,  be  converted 
by  the  ordinary  method. 

It  has  been  usual,  however,  to  apply  an  indirect  method, 
called  Conversion  by  Negation.  Take,  for  example,  "  Some  real- 
ities are  not  material  objects."  If  we  infer  that  "  Some  or  All 
material  objects  are  not  realities,"  we  commit  a  fallacy,  because 
the  predicate  of  the  converse  is  distributed,  while  it  is  not  dis- 
tributed as  subject  in  the  convertend,  because  the  proposition 
is  particular.  But  if  we  attach  the  negation  to  the  predicate 
in  the  original  we  have  "  Some  realities  are  not-material 
objects,"  or  "Some  realities  are  immaterial  objects."  The 
proposition  thus  becomes  I,  which  we  can  convert  simply  into 
"Some  immaterial  objects  are  realities."  This  proposition, 
then,  can  be  inferred  from  the  original,  and  the  process  of 
reaching  it  is  called  Conversion  b}T  Negation.  The  same  pro- 
cess is  applicable  to  similar  propositions.  Thus  "  Some  pleas- 
ant acts  are  not  just,"  would  become  "  Some  pleasant  acts  are 
not-just,  or  unjust,"  and  then  by  conversion,  "  Some  unjust 
acts  are  pleasant."  "  Some  men  are  not  agreeable  "  would  be 
converted  into  "  Some  disagreeable  persons  are  men." 

But  it  must  be  observed  that  the  quality  of  this  so-called 
converse  is  affirmative  while  the  convertend  is  negative,  and 
hence  the  process  of  conversion  by  negation  is  a  violation  of 
the  first  rule.  Besides,  we  have  been  led  by  it  to  affirm  some- 
thing positive  about  non-material  or  immaterial  objects  in  the 
assumed  converse,  when  the  convertend  merely  denies  some- 
thing about  material  objects.  ^Yhile  this  may  be  allowable  by 
some  other  process,  it  is  not  permissible  by  conversion.  The 
violation  of  the  first  rule  decides  that  matter.  Hence  we  con- 
clude that  proposition  O  is  really  not  convertible  at  all.  This 
is  the  general  opinion  of  logicians.  In  his  smaller  work  Jevons 
assumes  the  validity  of  the  process,  but  in  a  later  treatise  he 
concedes  his  error,  and  comes  over  to  the  general  view.  The 
proposition  which  is  supposed  to  be  the  converse  in  this  case 
is  really  the  result  of  a  double  process,  obversion  and  conver- 
sion, and  hence  is  the  converse  of  the  obverse,  or  the  contra- 
positive,  called  also  the  contraverse. 


160 


ELEMENTS  OF  LOGIC 


The  forms  of  Conversion  may  be  summarized  in  the  follow- 
ing table,  showing  also  the  impossible  forms  : 


Convertend. 

Converse. 

Impossible  Forms. 

All  S  is  P. 

Some  S  is  P. 

No  S  is  P. 

Some  S  is  not  P. 

A 
I 

E 

0 

Some  P  is  S. 

Some  P  is  S. 

No  P  is  S. 

Some  P  is  not  S. 

(None.) 

I 
I 

E 
0 

A,  E,  and  0. 

A,  E,  and  O. 

A  and  I. 

A,  E,  I,  and  0. 

3d,  Obversion. — This  is  sometimes  called  "Immediate 
Inference  by  Privative  Conception."  This  will  serve  as  a  good 
name  when  the  propositions  are  affirmative,  and  when  a  priva- 
tive term  can  be  found  for  the  purpose.  But  when  the  propo- 
sition is  negative,  and  when  a  privative  term  is  not  accessible, 
the  shorter  name  Obversion  is  much  to  be  preferred.  The  pro- 
cess consists  in  negating  the  copula  and  the  predicate  without 
conversion.  That  is,  the  quality  of  the  obvertend  must  be 
changed,  and  new  matter  introduced  into  the  obverse,  which 
shall  be  the  negative  concept  of  the  predicate  in  the  obvertend. 
Thus  the  proposition  "  All  men  are  mortal "  must  be  ob- 
verted  by  changing  the  quality  of  the  proposition  and  making 
the  predicate  of  the  obverse  the  negative  of  "mortal."  Hence 
the  obverse  of  it  will  be,  "  All  men  are  not  not-mortal,"  or  "  All 
men  are  not  immortal,"  or  "No  men  are  immortal."  Of  the 
proposition  "All  oaks  are  trees,"  it  will  be  "No  oaks  are  not 
trees."  It  will  be  apparent  that  this  is,  after  all,  identical  in 
meaning  with  the  original  proposition,  although  the  meaning 
is  stated  in  a  negative  tvay  for  the  sake  of  emphasis.  The  two 
negatives  together  make  an  affirmative,  and  we  can  have  a  neg- 
ative proposition  of  the  obverse  only  on  the  supposition  that 
one  of  the  negative  particles  qualifies  the  copula,  and  the  other 
the  predicate,  so  as  to  make  it  a  negative  conception.  In  some 
cases  this  negative  particle  can  be  joined  to  the  predicate,  in 
others  not. 

In  regard  to  the  negative  proposition  the  obversion  is  ac- 
complished simply  by  connecting  the  negative  particle  with 


IMMEDIATE  INFERENCE  161 

the  predicate,  which  both  changes  the  quality  of  the  proposi- 
tion and  the  character  of  the  predicate.  Thus  if  I  say  "  All 
men  are  not  angels,"  the  obverse  will  be  "All  men  are  not- 
angels."  A  better  illustration,  because  of  the  negative  term 
"unpleasant,"  is,  "All  pain  is  not  pleasant,"  the  obverse  of 
which  is  "All  pain  is  unpleasant,"  assuming  for  the  present 
that  "not  pleasant"  and  "unpleasant"  are  the  same.  This 
result  can  be  brought  about  by  the  more  complicated  process 
described  in  the  definition,  which  is  really  implied  in  the  case 
just  mentioned.  Thus,  if  I  say  "No  pain  is  pleasant,"  I  can 
obvert,  as  required  by  the  definition,  in  negating  the  cojDula 
and  predicate,  as  follows,  "No  pain  is  not  not-pleasant,"  or 
"No  pain  is  not  unpleasant."  But  this  is  very  awkward,  and 
as  two  negatives  make  an  affirmative  by  cancelling  each  other, 
we  assume  this  effect  and  have  either  the  original  proposition 
or  its  obverse,  as,  "  All  pain  is  unpleasant."  A  negative  propo- 
sition is  therefore  most  conveniently  obverted  by  transferring 
the  negative  particle  to  the  predicate,  which  changes  its  qual- 
ity. In  regard  to  obversion,  therefore,  it  is  noticeable  that  it 
can  be  applied  to  all  four  forms  of  propositions — A,  E,  I, 
and  O. 

4th.  Contraversion  or  Contraposition. — This  process  is 
usually  called  by  the  latter  name,  but  a  few  logicians  have  used 
the  former  term,  which  seems  decidedly  preferable  to  the  lat- 
ter, because  it  indicates  to  some  extent  by  the  very  name  the 
natm-e  of  the  process  involved.  Jevons,  using  the  term  Con- 
traposition, calls  it  a  form  of  conversion,  but  this  is  only 
partly  true,  because  the  conversion  takes  place  after  the  pro- 
cess of  obversion  has  been  performed,  as  will  be  shown  after 
defining  what  the  process  is. 

Contraversion,  or  Contraposition,  consists  in  the  negation  of 
the  copula  and  of  the  predicate  with  conversion.  That  is,  we 
negate  the  copula  or  proposition  and  the  predicate,  and  then 
convert,  It  amounts  to  the  same  thing  to  take  the  negative 
of  the  predicate  for  the  subject  of  the  contraverse,  and  deny 
the  connection  between  it  and  the  subject  of  the  contravertend 
if  the  contravertend  be  affirmative,  and  affirm  the  connection 
11 


162  ELEMENTS  OF  LOGIC 

if  the  contravertend  be  negative.  This  can  be  better  explained 
by  an  example.  Take  the  proposition  "All  men  are  mortal." 
By  the  very  terms  of  this  judgment  the  class  "men"  is  wholly 
included  in  the  class  "  mortal,"  as  in  Fig.  5  or  Fig.  14.  Hence 
it  is  necessarily  excluded  from  everything  outside  the  circle 
"  mortal."  I  can  therefore  affirm  that  "  All  men  are  not  in  the 
class  of  those  who  are  not  mortal ; "  or  more  briefly,  that  "  No 
men  are  immortal."  By  simple  conversion  of  E  we  get  "No 
immortals  are  men."  But  noticing  that  the  inclusion  of 
"men  "  among  the  "  mortal"  excludes  those  who  are  not  mor- 
tal from  the  class  "  men,"  I  may  as  well  affirm  that  fact  directly, 
and  hence  from  the  original  I  can  infer  at  once,  from  "  All 
men  are  mortal,"  that  "All  not-mortals  are  not  men."  The 
result  is  the  same  by  either  process,  and  hence  we  have  only 
to  apply  the  principle  to  any  similar  proposition.  Thus  "All 
oaks  are  trees,"  when  contraverted  will  be  "  All  not-trees  are 
not  oaks,"  etc.  We  seem  in  all  such  cases  to  reach  the  result 
without  any  roundabout  process.  But  nevertheless  that  of 
obversion  is  actually  involved,  and  it  was  the  failure  to  remark 
this  fact  which  has  led  Jevons  to  deny  contraversion  or  contra- 
position of  I  and  O,  and  admit  it  only  in  an  indirect  way  for  E. 
But  if  we  observe  that  obversion  is  implied  in  the  process, 
to  be  followed  by  the  conversion  of  the  obverse,  we  shall  find 
that  contraversion  is  applicable  to  E  and  O  as  well  as  to  A,  and 
in  the  same  manner,  but  is  not  applicable  to  I  because  its  ob- 
verse, O,  cannot  be  converted.  Thus  take  a  case  in  E,  "  No 
Caucasians  are  negroes,"  or  "  All  Caucasians  are  not  negroes." 
Obverting  this  according  to  rule  we  have,  "  All  Caucasians  are 
not-negroes,"  and  then  converting  we  have,  "  Some  not-negroes 
are  Caucasians."  Or,  "No  men  are  perfect."  Obverted  it  is 
"  All  men  are  not-perfect,"  and  converted,  "  Some  not-perfect 
(imperfect)  beings  are  men."  Then  again  take  O,  "  Some  men 
are  not  Americans."  Obverting  we  have,  "  Some  men  are  not- 
Americans,"  and  converting  we  have,  "Some  not-Americans  are 
men."  It  will  be  noticed  in  this  that  we  have  a  result  which 
has  been  called  Conversion  by  Negation,  as  already  explained. 
But  having  shown  that  O  cannot  in  reality  be  converted,  it  re- 


IMMEDIATE  INFERENCE  163 

mains  to  observe  that  those  who  have  assumed  that  it  could 
be  converted  in  this  indirect  manner  have  confused  the  pro- 
cess with  contraversion  or  contraposition.  O  is,  therefore, 
contravertible.     I  remains  to  be  considered. 

Take  the  proposition  "  Some  men  are  wise."  Obvert  it  and 
we  have  "  Some  men  are  not  not-wise  (foolish)."  Thus  we 
have  proposition  O,  "  Some  men  are  not  foolish,"  and  as  we 
have  seen  that  O  cannot  be  converted,  we  can  proceed  no 
further  with  the  case.  I,  therefore,  cannot  be  contraverted. 
A,  however,  is  contraverted  into  E  ;  E  into  I ;  O  into  I,  and  I 
not  at  all.  It  might  be  added,  also,  that  A  can  be  contra- 
verted into  0  as  well  as  E,  because  O  is  the  subalternate  of  E, 
and  so  is  equally  true  with  it.* 

*  The  first  thing  to  be  remarked  about  Contraversion  is,  that,  unlike 
Conversion,  the  process  involves  a  change  in  the  quality  of  the  judgment, 
a  change  from  the  affirmative  in  the  contravertend  to  the  negative  in  the 
contraverse,  and  from  the  negative  in  the  contravertend  to  the  affirmative 
in  the  contraverse.  The  accompaniment  of  this  fact  and  the  cause  of  it, 
perhaps,  is  the  second  incident,  which  is  that  there  is  a  change  of  matter 
as  well  as  form  in  the  proposition.  The  subject  in  the  contraverse  is  new 
matter,  and  we  may  well  question  the  validity  of  the  whole  process,  since 
immediate  inference  aims  to  deduce  from  a  content  already  given  noth- 
ing more  than  it  contains  ;  that  is,  immediate  inference  purports  to  give 
the  same  kind  of  matter,  not  always  the  same  quantity,  in  the  conclusion 
that  is  given  in  the  premise.  That  new  matter  appears  in  contraversion 
or  contraposition  is  evident  from  the  fact  that  in  the  original  proposition 
nothing  is  said  about  the  negative  of  the  predicate,  and  it  may  seem  gra- 
tuitous to  introduce  it  into  the  contraverse.  Thus,  ' '  All  men  are  mortal  " 
asserts  nothing  about  the  "  not-mortal,"  or  the  "immortal."  Why,  then, 
assert  anything  about  them  in  the  contraverse  ?  Again,  a  more  important 
objection  is  that  the  inference  can  be  at  best  only  formal,  because  we  do 
not  know  from  "  All  men  are  mortal  "  that  there  are  any  "  not-mortals." 
Undoubtedly  we  can  assert  that  "All  not-mortals  are  not  men,"  but  if 
we  suppose  that  the  contravertend  expresses  a  real  and  material  fact, 
does  it  follow  that  the  contraverse  also  expresses  a  real  fact  ?  If  so,  it 
would  imply  that  there  are  real  "  not-mortals,"  about  which,  as  a  fact, 
we  know  nothing.  Of  course,  if  there  are  any  beings  other  than  "mor- 
tal," the  contraverse  will  be  materially  true,  but  this  existence  of  them 
cannot  be  inferred  from  the  original  proposition.  In  immediate  inference 
the  conclusion  should  be  the  same  really  and  materially  as  the  premise  ; 
that  is,  it  should  be  materially  as  well  as  formally  true  if  the  premise  is 


164 


ELEMENTS  OF  LOGIC 


Conversion  and  Contraversion  may  be  compared  in  the  fol- 
lowing table,  and  it  will  be  seen  at  once  how  they  contrast 
with  each  other  : 


Original 
Proposition. 


All  S  is  P  .  A. 


Some  S  is  P  .  I. 
No  S  is  P  .  E. 


Some  S  is  not  P  .  O. 


Converse. 


Some  P  is  S  .  I. 


Some  P  is  S 
No  P.  is  S  . 
Some  P  is  not  S  . 
(None.) 


Impossi- 
ble. 


A.  E.  O. 


A.  E.  O. 
A.I. 


A,  E,  I,  O. 


Contraverse. 


No  P  is  not  S  - 
Some  not-P  is  not  S  , 

(None.) 


Some  not-P  is  S  .  I. 
Some  not-P  is  S  .  I. 


Impossi- 
ble. 


A,  I. 
A,  E,  I,  O. 


A,  E,  O. 
A,  E,  O. 


First,  it  is  noticeable  that  there  is  a  change  of  quality  in 
Contraversion.     Then  where  we  convert  A  into   I  only,   we 

both.  Of  course,  if  the  premise  is  only  formal,  or  consists  of  pure  con- 
ceptions, the  conclusion  cannot  assert  more.  But  this  only  shows  that 
we  cannot  go  beyond  the  matter  of  the  original  proposition.  What,  then, 
becomes  of  the  validity  of  contraversion  ? 

The  only  possible  reply  to  this  question  is,  that  we  are  always  assured 
by  the  law  of  contradiction  between  tbe  predicate  and  its  negative  that 
they  are  mutually  exclusive,  and  that  what  is  affirmed  of  one  can  be  de- 
nied of  the  other,  and  vice  versa  ;  and  second,  that  practically  there  are 
real  existences,  in  the  case  of  the  largest  number  of  propositions,  which 
belong  to  a  class  that  is  the  negative  of  the  predicate.  But  this  reply  is 
not  satisfactory,  because  the  law  of  contradiction,  as  mentioned,  can  only 
give  a  formal  inference  and  not  a  material  one,  and  because  the  principle 
does  not  justify  a  material  inference  where  other  objects  than  those  in 
the  predicate  are  not  known  to  exist ;  and  their  existence  is  not  implied 
by  the  material  truth  of  the  original.  However,  it  is  to  be  remarked 
that,  if  any  objects  exist  which  are  excluded  from  the  predicate  of  the 
original  proposition,  the  inference  will  then  be  materially  true,  and  that 
this  inference  is  contingent  upon  that  condition,  including,  on  the  one 
hand,  the  existence  of  certain  objects  other  than  the  predicate,  and,  on 
the  other,  the  exclusion  or  contradiction  between  them  and  the  given 
predicate.  We,  therefore,  conclude  from  the  existence  of  this  condition 
that  contraversion  is,  at  least  materially  considered,  a  conditional  infer- 
ence ;  not  conditioned  upon  the  material  truth  of  the  contravertend,  but 
upon  the  material  existence  of  data  that  are  excluded  from  the  predicate 
of  the  original,  or  contravertend.  In  conversion  and  obversion  the  con- 
verse and  obverse  are  always  materially  the  same  in  kind  as  the  proposi- 
tions from  which  they  are  drawn,  so  that,  if  the  originals  are  materially 
true,  the  derivatives  are  also.  But  in  contraversion,  as  we  have  seen, 
the  material  truth  of  the  contraverse  is  not  dedncible  from  anything  act- 


IMMEDIATE  INFERENCE  165 

can  contra  vert  it  into  both  E  and  0,  as  we  can  convert  E 
into  both  E  and  0,  and  contravert  it  into  I  only.  Again,  as  I 
can  be  converted  into  I,  and  O  cannot  be  converted  at  all,  so 
I  cannot  be  contraverted,  and  O  can  be  into  I.  The  impossible 
cases  denote  that  the  forms  included  under  that  head  cannot 
be  derived  from  the  original  proposition  by  the  process  indi- 
cated. 

In  the  practical  application  of  Contraversion  we  must  be 
careful  not  to  confuse  nego-positive  with  infinitated  negative 
conceptions.  Thus  if  we  say  that  "  All  just  acts  are  expedient," 
the  contraverse  or  contrapositive  is,  "  All  not-expedient  acts 
are  not  just."  But  we  are  not  entitled  to  say  "All  not-expedient 
acts  are  unjust,"  because  all  that  we  know  from  the  original 
assertion  is  that  they  are  excluded  from  the  "  just,"  and  the 
negative  of  this,  which  is  " not-just,"  includes  the  "unjust," 
and  those  which  are  neither  just  nor  unjust,  namely,  indif- 
ferent acts,  such  as  physical  actions,  and  we  do  not  know  from 
the  original  whether  the  "  not-expedient "  acts  are  excluded 
from  those  which  are  neither  just  nor  unjust,  or  included  in 
the  "  unjust."  We  only  know  that  they  are  excluded  from  the 
"just."  Besides,  to  say  that  "All  not-expedient  acts  are  un- 
just "  is  to  produce  a  proposition  in  A,  when  the  contraverse 
should  be  E.  The  conception  "unjust  "  is  not  necessarily  co- 
extensive with  the  negative  "  not- just,"  which  is  infinitated. 
If  it  were,  there  would  be  no  objection  to  their  identification, 
and  then  the  original  contraverse,  "  All  not-exjDedient  acts  are 

ually  affirmed  in  the  contravertend,  but  is  conditioned  solely  upon  the 
existence  of  objects  other  than  the  predicate  of  the  latter.  It  may  then 
be  regarded  as  formally  true,  but  materially  indeterminate.  The  same 
will  be  true  of  Inversion,  of  which  we  have  yet  to  speak,  and  would  be 
true  of  Obversion  except  for  the  fact  that  it  expresses  merely  in  a  negative 
way  the  same  as  the  original  proposition.  In  Conversion,  however,  the 
converse  is  both  formally  and  materially  true  ;  that  is,  materially  true 
when  the  convertend  is  so.  The  following  is  a  classification  of  immediate 
inferences  based  upon  the  above  considerations  : 

Immediate  {Formally  a^d  materi^y  true]  g^^n" 

inference   |Formally  trUG)  but  materially  illdeterminate  j  ^^^^ 


160  ELEMENTS  OF  LOGIC 

not  just,"  could  be  obverled  into  "  All  not-expedient  acts  are 
unjust  "  (not-just).  But  only  on  that  condition  can  it  be 
done.  The  same  remarks  apply  to  the  substitution  of  inex- 
pedient for  "not-expedient."  A  better  and  more  self-evident 
illustration  of  what  is  here  maintained  can  be  seen  in  the 
proposition  "  All  human  actions  are  free,"  the  contraverse  of 
which  must  be,  "  All  not-free  actions  are  not  human,"  not  "  All 
not-free  actions  are  inhuman."  Or  again,  "  Human  kindness 
is  a  virtue,"  the  contraverse  of  which  is  "  All  that  is  not  virtue 
is  not  human  kindness,"  not  "  All  that  is  not- virtue  is  in- 
human kindness."  The  absurdity  of  the  last  is  palpable,  but 
it  is  precisely  what  we  have  to  guard  against. 

5th.  Inversion. — Inversion  is  not  often  treated  by  logicians, 
and  it  is  not  a  process  of  any  practical  importance.  Besides, 
it  has  first  to  be  obtained  through  such  combinations  of  the 
three  previous  processes  as  not  to  be  apparent  at  first  sight. 
But  the  fact  that  it  is  possible  requires  mention  in  a  com- 
plete treatment  of  immediate  inferences.  Inversion  is  the 
process  of  inferring  from  one  proposition  another  which  shall 
contain  for  its  subject  the  negative  of  the  subject  in  the  orig- 
inal, and  for  its  predicate  the  predicate  of  the  original.  The 
result  can  be  obtained  in  more  than  one  way.  We  may  alter- 
nate conversion  and  obversion,  or  use  contraversion  with 
either  or  both  the  other  processes.  As  contraversion,  how- 
ever, is  merely  a  combination  of  conversion  and  obversion,  we 
may  use  only  the  former  method.  We  may  begin  with  either 
conversion  or  obversion. 

Take  the  proposition  "  All  horses  are  animals."  Convert 
and  we  have  "  Some  animals  are  horses."  Then  obvert  this 
result,  which  gives  "  Some  animals  are  not-horses."  Here 
comes  the  turn  for  conversion,  but  as  the  proposition  is  O  we 
can  proceed  no  further.  But  if  we  begin  with  obversion  the 
first  transformation  gives  us  "  No  horses  are  not-animals," 
then  convert  and  we  have  "  All  not-animals  are  not  horses." 
Obvert  again  and  we  have  "  All  not-animals  are  not-horses  " 
(remembering  that  obversion  is  only  a  change  of  quality  in  the 
proposition),  and  then  converting  this  obverse  we  have  "  Some 


IM MEDIA  TE  INFERENCE 


167 


not-horses  are  not-aniraals."  Obverting  this  again  we  have 
"Some  not-horses  are  not  animals,"  which  is  the  required 
proposition,  having  the  negative  of  the  original  subject  for  its 
subject,  and  the  original  predicate  for  its  predicate. 

If  we  take  proposition  E,  "No  men  are  quadrupeds,"  and 
begin  with  conversion,  we  have  "  No  quadrupeds  are  men." 
Obvert  this  and  we  get  "  All  quadrupeds  are  not-men,"  and 
applying  conversion  we  have  "  Some  not-men  are  quadru- 
peds," the  required  inverse.  If  we  begin  with  obversion  we 
shall  find  that  the  result  cannot  be  obtained. 

Without  going  through  the  process  with  I  and  O,  which  the 
student  can  do  for  himself,  we  shall  content  ourselves  with 
the  assertion  that  they  cannot  be  inverted.  A  and  E  are  the 
only  two  invertible,  A  into  O,  and  E  into  I.  It  must  be  re- 
marked, however,  that  the  material  truth  of  the  conclusion  in 
either  case  is  dependent  upon  the  same  conditions  as  in  con- 
traversion.  Take  the  case  of  the  inversion  of  A  into  the  form 
"  Some  not-S  is  not  P,"  from  "  All  S  is  P,"  and  it  is  clear 
from  "All  S  is  P  "  that,  so  far  as  we  know,  all  things  outside  of 
S  are  included  in  P,  and  if  so,  we  could  hardly  conclude  that 
some  of  them  were  not  so  included,  and  hence  "  not  P."  But 
if  there  are  any  objects  outside  of  P,  then  it  is  clear  from  the 
subalternate  of  the  contraverse  of  "  All  S  is  P,"  that  "  Some 
not-S  is  not  P."  But  it  can  be  materially  true  only  on  that 
condition.  Hence  I  have  placed  inversion  in  the  class  of  in- 
ferences which  are  materially  indeterminate.  It  has  no  prac- 
tical importance,  however,  and  is  applicable  only  to  A  and  E, 
and  not  to  I  and  O.  It  should  be  observed  that  it  involves  a 
change  of  quality  in  the  deduced  proposition.  All  four  pro- 
cesses can  be  compared  in  the  following  table  : 


Original. 


All  S  is  P  .  A. 
No  S  is  P  .  E. 


Some  S  is  P  .  I. 
Some  S  is  not  P  .  O. 


Converse. 


Some  P  is  S  .  I. 


No  P  is  S  , 
Some  P  is  not  S  , 
Some  P  is  S 
(None.) 


Contraverse. 


All  not  P  is  not  S  . 
Some  not  Pis  not  S. 

Some  not  P  is  S 

(None. ) 
Some  not  P  is  S 


Inverse. 


Some  not-S  is  not  P  .  O. 

Some  not-S  is  P  .  I. 
(None.) 
(None.) 


168  ELEMENTS  OF  LOGIO 

6th.  Contribution. — I  have  called  two  processes  of  imme- 
diate inference  by  this  name  because  the  same  general  princi- 
ple is  involved  in  both.  They  are  called  Immediate  Inference 
by  added  Determinants,  and  Immediate  Inference  by  Complex 
Conceptions.  The  essential  characteristic  in  common  between 
them  is  that  what  is  affixed  to  the  subject  as  a  modifier  may  also 
be  affixed  to  the  predicate  in  the  same  sense.  What  is  contri- 
buted to  one  may  be  contributed  to  the  other.  Its  simplest 
illustration  is  in  mathematics.     Thus  if  x  =  a,  x  +  1  =  a  +  1. 

(a)  Inference  by  Added  Determinants.  —  This  consists  in 
merely  adding  some  adjective  or  similar  term  to  both  subject 
and  predicate.  Thus  if  "  A  brick  house  is  a  dwelling,"  "  A 
good  brick  house  is  a  good  dwelling."  There  is  one  important 
limitation  to  this  process,  however,  and  it  is  that  the  addition 
must  represent  the  same  quality  and  quantity  to  both  subject 
and  predicate.  Thus,  "  Dogs  are  quadrupeds  "  may  be  modi- 
fied by  adding  "  useful "  to  both  terms,  but  not  by  saying 
"  The  largest  dogs  are  large  quadrupeds."  Nor  can  we  infer 
from  "  Dwarfs  are  men,"  or  "  A  cottage  is  a  building,"  that 
"  Large  dwarfs  are  large  men,"  or  "  A  large  cottage  is  a  large 
building."  The  reason  for  the  error  in  these  cases  is  that 
the  addition  of  quantity  in  each  case  is  not  the  same.  When 
the  superlative  degree  is  added  it  implies  a  comparison  be- 
tween the  special  individuals  mentioned  or  described  and  the 
whole  class  which  is  thus  limited,  so  that  the  same  addition  is 
not  made  to  subject  and  predicate.  It  is  the  same  with  such 
terms  as  "large,"  "long,"  "small,"  "short,"  etc.,  in  certain 
cases,  namely,  when  an  unequal  comparison  is  suggested. 
Terms,  therefore,  expressing  quantity  must  be  used  carefully 
in  this  form  of  inference.  Terms  expressing  quality,  however, 
can  be  used  with  perfect  freedom,  provided  they  are  not  used 
equivocally. 

(b)  Inference  by  Complex  Conception. — This  consists  in  the 
addition  of  words,  phrases,  or  clauses  that  make  the  subject 
and  predicate  complex  in  their  nature.  Thus  from  "  Apples 
are  fruit "  I  may  infer  that  "A  box  of  apples  is  a  box  of  fruit," 
or  from  "  Pigeons  are  birds,"  "  A  flock  of  white  pigeons  is  a 


IMMEDIATE  INFERENCE  109 

flock  of  white  birds,"  etc.  But  here  again  we  have  to  be 
warned  against  the  addition  of  expressions  which  indicate  the 
addition  of  a  greater  quantity  to  one  side  than  to  the  other. 
Thus  we  cannot  infer  from  "Voters  are  men,"  that  "The  ma- 
jority of  voters  is  the  majority  of  men."  The  same  principle 
holds  here  as  in  inference  by  added  determinants. 

Inferences  by  contribution  have  little  importance,  as  they 
are  seldom  employed,  and  if  fallacies  occur  in  this  use  they 
are  so  easily  detected  and  corrected  that  little  provision  re- 
quires  to  be  made  against  their  occurrence. 

7th.  Antithesis. — There  is  a  very  common  error  in  dealing 
with  propositions,  which  it  is  necessary  to  remark.  "We  have 
seen  that  an  exclusive  proposition  implies  a  complementary 
opposite,  and  it  frequently  occurs  that  we  infer  from  an  or- 
dinary universal,  A  or  E,  proposition  one  which  is  its  opposite 
both  in  regard  to  the  subject  and  predicate.  Thus,  if  we  assert 
that  "  All  white  objects  are  visible,"  many  persons  think  them- 
selves entitled  to  infer  that  "All  not-white  objects  are  not 
visible ; "  or  from  "  All  good  men  are  wise,"  that  "  All  bad 
men  are  not  wise."  But  such  an  interpretation  and  inference 
is  a  case  of  illicit  distribution  of  the  predicate.  If  the  propo- 
sition were  either  exclusive  or  a  definition,  the  inference  might 
be  drawn.  Formally,  however,  no  one  ever  has  the  right  so 
to  treat  it.  In  ordinary  antithesis  this  relation  is  assumed,  as 
is  so  frequent  in  the  Book  of  Proverbs  ;  for  example,  "  In  the 
multitude  of  words  there  wanteth  not  transgression  ;  but  he 
that  refraineth  his  lips  doeth  wisely."  And  in  all  ordinary 
cases  where  subject  and  predicate  are  coterminous,  the  same 
can  be  done.  In  such  instances  we  are  entitled  to  infer  the 
complementary  opposite.  But  we  should  not  generalize  a 
privilege  of  this  kind  and  apply  it  to  all  universal  propositions. 
In  the  universal  propositions  as  defined,  we  have  no  data  for 
asserting  anything  outside  the  subject,  unless  we  assume 
something  to  exist  outside  the  predicate,  which  we  have  no 
right  to  assume  from  the  given  proposition.  But  if  it  be 
known  or  assumed  that  something  exists  outside  the  predicate, 
we  may  then  infer  a  contrapositive  or  the  contraverse,  and  the 


170  ELEMENTS  OF  LOGIO 

limited  converse  of  this  which  has  been  shown  to  be  the  inverse 
of  the  original  proposition.  Any  other  inference  than  those 
based  upon  this  supposition,  is  an  illicit  assumption  of  distri- 
bution of  the  predicate  when  the  original  proposition  is  affirm- 
ative, and  an  illicit  distribution  of  the  negative  of  the  subject 
when  the  proposition  is  negative.  Thus,  in  the  first  case,  if 
we  infer  from  "All  men  are  mortal  "  that  "All  not  men  are  not 
mortal,"  we  assume  a  distribution  of  the  predicate  which  is  not 
stated  in  the  original.  In  the  second  case,  if  we  infer  from 
"  No  men  are  quadrupeds  "  that  "  No  not-men  are  quadru- 
peds," we  assume  a  distribution  of  the  negative  of  the  subject 
"  men."  This  cannot  be  true,  because  a  part  of  the  things  ex- 
cluded from  "  men  "  is  included  in  "  quadrupeds."  This  error 
majr  be  called  illicit  contrast,  or  illicit  antithesis.* 

*  References  on  Immediate  Inference  are  the  following:  Keynes:  For- 
mal Logic,  Part  II.,  Chaps.  III.,  IV.  and  V.;  Jevons  :  Studies  in  Deductive 
Logic,  Chap.  V. 


CHAPTER  XL 

PRINCIPLES   OF   MEDIATE   REASONING 

1st.  Definition. — Mediate  inference,  as  already  defined,  is 
reasoning  by  means  of  a  middle  term.  A  middle  term  is  one 
which  is  compared  with  two  otliers,  and  on  the  ground  of 
which  a  connection  or  relation  can  be  established  between  the 
two  terms  thus  compared  with  it.  Suppose  I  wish  to  prove 
that  "  Machines  are  useful."  This  is  the  conclusion,  and  its 
proof  will  depend  upon  two  other  propositions  containing  the 
two  terms  in  the  conclusion  and  comparing  them  with  a  mid- 
dle term.  If,  therefore,  I  can  obtain  assent,  first,  to  the  prop- 
osition  that  "  Mechanical  arrangements  for  the  application  of 
power  are  useful,"  and  second,  to  the  proposition  that  "Ma- 
chines are  mechanical  arrangements  for  the  application  of 
power,"  the  conclusion  will  follow  as  a  matter  of  course.  The 
terms  "machines"  and  "  useful  "are  compared  with  the  single 
term  "  mechanical  arrangements,"  etc.,  and  found  to  agree 
with  it,  and  so  it  must  be  inferred  that  they  agree  with  each 
other.  The  conclusion  is  thus  mediated  by  a  middle  term. 
The  reasoning  involved  in  this  process  is  called  the  Syllogism, 
or  Syllogistic  Reasoning.  It  differs  from  immediate  inference 
in  requiring  more  than  one  proposition,  and  more  than  two 
terms  to  start  from.  The  two  rules  which  determine  or  con- 
dition the  character  of  the  syllogism  are  : 

(a)  Every  syllogism  should  have  three,  and  only  three  terms. 

(b)  Every  syllogism  should  have  three,  and  only  three  proposi- 

tions. 
The  three  terms  are  called  the  Major,  the  Middle,  and  the 
Minor  terms.  Of  the  three  propositions  two  are  called  the 
Premises,  and  one  the  Conclusion.  The  Major  term  is  the 
predicate  of  the  conclusion  ;  the  Minor  term  is  the  subject  of 
the  conclusion.     The  Middle  term  is  found  only  in  the  prem- 


172  ELEMENTS  OF  LOGIC 

ises,  and  may  be  either  the  subject  or  the  predicate  as  the 
case  requires.  The  Major  Premise  contains  the  Major  and 
Middle  terms,  and  the  Minor  Premise  the  Minor  and  Middle 
terms.  Without  the  conclusion  being  expressed  there  is  no 
absolute  rule  for  determining  which  of  the  premises  is  the 
major  and  which  the  minor.  We  are  at  liberty  to  choose  either 
of  them  and  to  try  the  consequences  by  the  laws  of  the  syllo- 
gism. The  usual  method  of  the  logician  is  to  state  the  major 
premise  first,  the  minor  premise  second,  and  the  conclusion  last. 
But  in  common  discourse  and  reasoning  the  minor  premise 
may  come  first,  and  the  major  second  ;  or  the  conclusion  may 
even  come  first,  and  the  two  premises  after  it,  as  reasons  for 
its  truth.  As  an  illustration  of  a  syllogism  having  the  minor 
premise  first,  the  following  is  an  instance  : 

The  earth  revolves  around  the  sun. 

Bodies  revolving  around  the  sun  are  planets. 

Therefore  the  earth  is  a  planet. 
It  is  evident  from  the  conclusion,   and  from  what  has  been 
said  about  the  terms  which  it  contains,  that  the  minor  premise 
is  the  first  and  the  major  premise  the  second.     As  a  case  of 
the  conclusion  being  stated  first,  the  following  is  an  instance  : 

Comets  consist  of  matter,  for  they  obey  the  law  of  gravita- 
tion, and  whatever  obeys  the  law  of  gravitation  is  matter. 

The  symbols  which  we  shall  employ  for  the  several  terms  of 
the  syllogism  are  the  letters  S,  M,  and  P,  with  a  slight  modifi- 
cation of  their  previous  usage,  but  yet  essentially  the  same. 
S  shall  stand  for  the  minor  term,  which  is  the  subject  of  the 
conclusion,  and  P  for  the  major  term,  which  is  the  predicate  of 
the  conclusion.  They  thus  stand  respectively  for  subject  and 
predicate,  as  heretofore,  but  only  in  the  conclusion,  since  S  or  the 
minor  term  is  not  always  the  subject  in  the  minor  premise, 
and  P,  the  major  term,  is  not  always  the  predicate  of  the  major 
premise.  M  will  stand  for  the  middle  term.  The  combination 
of  these  terms  will  represent  the  premises  and  conclusion. 

Terms.  Propositions. 

M  =  Middle  term.  M  is  P  =  Major  Premise. 

S  =  Minor  term  =  Subject  of  Conclusion.  S  is  M  =  Minor  Premise. 

P  =  Major  term  =  Predicate  of  Conclusion.  S  is  P  =  Conclusion. 


PRINCIPLES  OF  MEDIATE  REASONING  173 

2d.  Rules  of  the  Syllogism. — The  rules  of  the  syllo- 
gism are  classified  in  slightly  different  forms,  but  substan- 
tially in  the  same  manner.  It  is  usual  to  include  among  them 
the  two  which  I  have  already  mentioned,  and  to  include  in  one 
of  them  a  statement  about  the  ambiguous  middle.  But  I  pre- 
fer to  specify  two  classes  of  rules  affecting  different  aspects  of 
the  syllogism.  The  first  two  already  mentioned,  and  with 
them  a  precaution  against  an  ambiguous  middle,  relate  to  the 
logical  matter  of  the  syllogism.  The  remaining  six  relate  to 
the  quantity  and  quality  of  propositions,  and  the  distribution 
of  terms,  as  effecting  the  conclusion.  We  shall  classify  them 
according  to  these  divisions. 

(A)  Eules  affecting  the  logical  matter  of  the  syllogism. 

1.  Every  syllogism  must  have  three  terms,  and  only 

three. 

2.  Every  syllogism  must  have  three  propositions,  and 

only  three. 

3.  No  term  shall  be  used  ambiguously. 

The  violation  of  the  first  of  these  rules  gives  rise  to  the  fal- 
lacy known  as  Quaternio  Terminorum,  or  the  Fallacy  of  Four 
Terms.  The  third  rule  is  a  corollary  of  the  first,  because  an 
ambiguous  term  is  a  word  with  two  meanings,  and  hence  is 
equivalent  to  two  terms,  so  that  it  is  only  a  form  of  introduc- 
ing four  terms  into  the  syllogism,  producing  in  effect  a  fallacy 
of  Quaternio  Terminorum. 

(B)  Kules  affecting  the  quantity  and  quality  of  propositions 

and  the  distribution  of  terms. 

4.  The  middle  term  must  be  distributed  at  least  once 

in  the  premises. 

5.  No  term  must   be    distributed  in  the  conclusion 

which  was  not  distributed  in  at  least  one  of  the 
premises. 

6.  No  conclusion  can  be  inferred  when  both  premises 

are  negative. 

7.  If  one  premise  be  negative  the  conclusion  must  be 

negative  and,  vice  versa,  in  order  to  prove  a  negative 
conclusion  one  of  the  premises  must  be  negative. 


174  ELEMENTS  OF  LOGIC 

8.  No  conclusion  can  be  drawn  when  both  premises 

are  particular. 

9.  If  one  of  the  premises  be    particular  the   conclu- 

sion must  be  particular. 

The  last  two  are  frequently  considered  as  corollaries  of  the 
preceding.  All  of  them  are  so  important  that  they  should  be 
thoroughly  committed  to  memory  by  the  student  for  constant 
reference. 

3d.  Symbolization  of  the  Syllogism  and  Representa- 
tion of  Formal  Fallacies. — The  syllogism  can  be  symbol- 
ized in  the  same  manner  as  propositions,  except  that  it  re- 
quires a  larger  number  of  circles  for  each  syllogism,  because 
there  are  three  terms  and  three  propositions.  The  represen- 
tation depends  upon  the  quantitative  relation  between  subject 
and  predicate,  and  can  be  used  to  indicate  whether  any  two 
terms  agree  or  exclude  each  other.  When  the  relation  be- 
tween subject  and  predicate  is  properly  observed,  and  when 
the  proper  distribution  of  terms  is  made,  the  conclusion  will 
be  valid.  But  if  the  middle  term  be  not  distributed  in  one  of 
the  premises,  or  if  the  major  or  minor  terms  be  distributed  in 
the  conclusion  when  they  are  not  distributed  in  the  prem- 
ises, a  fallacy  occurs,  which  is  called  a  Formal  Fallacy, 
because  it  is  a  violation  of  the  formal  principles  of  the  syllo- 
gism. Hence  such  reasoning  will  be  formally  invalid.  The 
conclusion  may  happen  to  be  materially  true,  but  under  the  cir- 
cumstances it  has  not  been  correctly  obtained  or  deduced 
from  the  premises,  as  a  conclusion.  These  formal  fallacies  are 
called  Illicit  Processes,  respectively,  of  the  major,  middle,  and 
minor  terms.  If  the  major  term  be  distributed  in  the  conclu- 
sion and  is  not  distributed  in  the  premise,  the  inference  is 
an  Illicit  Process  of  the  Major  Term.  If  the  middle  term  be 
not  distributed  at  least  once  in  the  premises,  the  inference  is 
an  Illicit  Process  of  the  Middle  Term.  If  the  minor  term  be 
distributed  in  the  conclusion  and  is  not  distributed  in  the 
premise,  the  inference  is  an  Illicit  Process  of  the  Minor  Term. 

There  are  two  ways  in  which  we  can  represent  symbolically 
the  valid  and  the  illicit  processes  of  formal  inference.     We 


PRINCIPLES  OF  MEDIATE  REASONING  175 

shall  employ  both  of  them.  We  must  remember  the  distinc- 
tion, however,  between  the  formal  truth  or  falsehood  of  the  in- 
ference and  the  material  truth  or  falsehood  of  the  propositions. 
The  two  may  have  no  relation  to  each  other.  Thus  we  may 
reason  correctly  from  false  premises,  or  falsely  from  correct 
premises.  Hence,  as  we  are  dealing  here  only  with  the  manner 
of  getting  the  conclusion,  we  may  set  aside  all  questions  about 
the  nature  of  the  premises  materially  considered.  The  stu- 
dent must  always  be  prepared  to  detect  the  difference  between 
the  truth  of  the  conclusion  and  the  validity  or  invalidity  of  the 
process  by  which  we  obtain  it.  It  is  a  common  fault  of  our 
minds  to  accept  the  reasoning  if  we  accept  the  conclusion, 
and  to  impeach  the  reasoning  if  we  do  not  accept  the  conclu- 
sion. But  this  is  a  fallacy  of  itself,  and  we  must  dispel  such 
assumptions  from  our  minds,  or  we  shall  not  be  able  in  all 
cases  to  detect  the  true  or  false  character  of  the  reasoning. 
On  the  other  hand,  we  must  not  be  induced  by  the  purely 
formal  correctness  of  the  reasoning  to  accept  the  material 
truth  of  the  conclusion  on  that  account.  The  most  perfect 
reasoning  is  that  in  which  the  premises  are  materially  correct 
and  the  reasoning  formally  correct.  This  will  give  a  conclu- 
sion which  is  both  formally  and  materially  valid.  Now  to 
illustrate  only  the  formal  process,  and  to  symbolize  it  accord- 
ingly, we  shall  take  first  a  valid  case,  the  ordinary  syllogism  : 
thus  "Metals  are  elements,"  "Iron  is  a  metal,"  "Therefore 
iron  is  an  element,"  will  be  represented  by  the  circles  in  Fig. 
18,  and  the  distribution  of  terms  in  the  accompanying  diagram. 

®  =  P 
®  =  M 

®  =  P 

Fig.  18.  Representation  of  Distribution. 

In  Fig.  18,  "  metals  "  are  represented  as  included  in  the  class 
"elements,"  and  "iron"  in  the  class  "metals."  It  must, 
therefore,  follow  that  "  iron "  is  included  in  the  class  "  ele- 
ments," as  the  lesser  is  included  in  the  greater,  or  the  part  in 


176  ELEMENTS  OF  LOGIC 

the  whole.  The  diagram  shows  that  the  middle  term  has  been 
distributed  at  least  once,  and  that  no  term  has  been  distrib- 
uted in  the  conclusion  which  was  not  distributed  in  the  prem- 
ises. The  propositions  are,  of  course,  affirmative.  But  we 
may  take  a  case  where  one  of  them  is  negative,  and  so  have 
an  instance  of  exclusion.  Thus,  "Iron  is  a  metal,"  and 
"  Wood  is  not  a  metal."  Therefore  "  Wood  is  not  iron."  This 
syllogism  will  be  represented  also  by  three  figures,  but  one  of 
them  is  excluded  from  the  other  two,  as  required  by  the  nega- 
tive jDrojDOsition. 

©  =M 
®  X© 

©X® 

Fig.  19.  Representation  of  Distribution. 

Fig.  19  represents  "  iron  "  as  included  in  class  "  metal,"  and 
"wood"  as  excluded  from  the  same  class,  so  that  it  must  be 
excluded  from  all  contained  within  the  class  "  metal."  The 
diagram  shows  the  observation  of  the  rules  in  another  form. 

Instead  of  a  dotted  line  we  might  use  a  smaller  circle  in 
either  case,  as  is  usual,  and  perhaps  better,  in  most  instances. 
When  the  predicate  is  not  exhausted  in  the  subject  the  com- 
plete smaller  circle  is  much  more  convenient,  and  hence  in  the 
future  it  will  be  employed,  especially  in  testing  the  forms  of 
illicit  process. 

Had  it  been  used  in  the  previous  symbols  Fig.  18  would 
stand  as  Fig.  20,  and  Fig.  19  as  Fig.  21. 


Fig.   20.  Fig.  21. 


The  student  or  the  teacher  may  vary  the  forms  of  figures  to 
suit  other  valid  modes  of  reasoning;.     We  turn  next  to  test 


PRINCIPLES  OF  MEDIATE  REASONING  177 

and  represent  the  forms  of  illicit  process.  Take  first  the  case 
of  illicit  middle.  It  occurs  in  the  following  syllogism,  "Men 
are  mortal."  "  Horses  are  mortal "  "  Therefore  horses  are 
men."    It  is  represented  in  Fig  22. 


U  Men    )\Horses)j 


®  =  M 
(D  =  M 


Fig.  22.  Representation  of  Distribution. 

As  both  propositions  of  the  premises  are  affirmative,  the 
predicate,  which  in  this  case  is  the  middle  term,  is  not  dis- 
tributed, and  hence  an  attempt  to  draw  the  conclusion, 
"  Horses  are  men,"  or  S  is  P,  is  illicit  or  invalid,  because  a  vio- 
lation of  Rule  4.  It  is  represented  by  Fig.  22,  in  that  both 
"horses"  and  "men"  are  included  in  the  class  mortal,  but  we 
have  no  right  to  infer  therefrom  that  "  horses "  are  included 
in  the  class  "  men."  The  fact  that  we  know  it  is  not  the  case 
helps  us  to  perceive  the  impossibility  of  it.  But  even  in  cases 
where  the  conclusion  would  be  true  as  a  fact,  we  should  have 
no  right  to  draw  it  in  this  manner.  Thus  we  may  say,  "  All 
Americans  are  men,"  and  "  All  Virginians  are  men,"  but  we 
should  have  no  right  to  infer  that  "  All  Virginians  are  Ameri- 
cans," although  it  is  true  as  a  matter  of  fact.  The  fallacy  here 
is  that  of  an  Illicit  middle  term. 

Illicit  process  of   the  major  may  occur  as  follows,  and  is 
likely  to  be  a  very  frequent  fallacy.     If  from  "  Men  are  mor- 
tal," and  "Horses  are  not  men,"  we  infer  that  "Horses  are  not 
mortal,"  we  have  more  in  our  conclusion  than  is     ^^ 
given  in  the  premises.    Fig.  22  will  also  represent     k^J  ~ 
this  form.     But  the  diagram  for  distribution  of     (g)  y  (m) 
terms  will  be  different.     It  will  be  as  follows,  and 
represents  the  major  term  as  distributed  in  the     (Sj  X  (P) 
conclusion  when  it  is  not  distributed  in  the  prem-    Representation 

*  of  Distribution. 

ises.     The  major  premise  is  affirmative,  and  hence 
the  predicate,  which  in  this  case  is  the  major  term,  is  not  dis- 
12 


178  ELEMENTS  OF  LOGIC 

tributed.  But  the  minor  premise  being  negative,  the  conclu- 
sion, according  to  Rule  7,  must  be  negative.  We  have  seen  that 
all  negative  proj^ositions  distribute  the  predicate,  and  as  this 
is  the  major  term  in  the  conclusion,  it  must  be  distributed,  if 
drawn  at  all,  although  it  is  not  distributed  in  the  major  prem- 
ise. The  error  is,  therefore,  an  illicit  process  of  the  major  term. 
Fig.  22  represents  it  by  showing  that  the  class  "  horses  "  is 
excluded  from  the  class  "  men,"  as  asserted  by  the  minor 
premise,  and  yet  is  not  excluded  from  the  class  "  mortal,"  as 
the  conclusion  would  indicate.  The  reasoning  is  therefore  a 
violation  of  Eule  5. 

Illicit  process  of  the  minor  term  is  quite  as  easily  repre- 
sented. It  occurs  in  the  following  syllogism:  "All  horses 
are  quadrupeds,"  and  "  All  quadrupeds  are  animals."  "  There- 
fore all  animals  are  horses."     It  is  represented  in  Fig.  23. 


(P)  =  M 

®  =  S 

®  =  P 

Fig.  23.  Kepresentation  of  Distribution. 

In  the  minor  premise  the  predicate  S,  or  "animals,"  is  not 
distributed,  because  the  proposition  is  affirmative,  but  in  the 
conclusion  it  is  distributed,  because  the  proposition  is  univer- 
sal, and  thus  to  distribute  it  is  violation  of  Eule  5.  The  im- 
possibility of  the  conclusion  is  evident  in  the  representation, 
on  the  ground  that  the  larger  class  cannot  be  included  in  the 
smaller,  or  the  whole  in  the  part,  as  it  is  represented  in  the 
conclusion  actually  drawn.  Had  we  said  "  Some  animals  are 
horses,"  the  conclusion  would  have  been  valid  as  conform- 
ing to  all  the  rides.  But  the  minor  term  is  distributed  in 
the  conclusion  first  drawn,  and  is  not  distributed  in  the 
premise. 

Tbe  violation  of  Rule  6  occurs  in  the  following  syllogism, 
and  is  represented  by  either  of  Figs.  24  and  25,  "  No  solids 


PRINCIPLES  OF  MEDIATE  REASONING  179 

are  liquids,"  and  "No  metals  are  liquids."     "Therefore  no 
metals  are  solids,"  or  "  Metals  are  solids." 


j       Solids        )  (       Metals      \ 


Fig.  24. 


Fig.  25. 


Fig.  24  represents   the  premises  as    /rj\  x  (™\ 
negative  propositions,  in  which  the  sub- 
jects "solids "and  "metals "are  both    (S)  X  (M) 
excluded  from  "liquids,"  and  excluded    /~n  y  fjTjS       /q\  _  -jy- 
from  each  other.     But  their  common    ^^^      ^"^       ^^^ 
exclusion  from  the  middle  term  does       Diagram  o£  Distributi™- 
not  necessarily  imply  their  exclusion  from  each  other,  as  is  ap- 
parent in  Fig.  25,  where  they  are  both  excluded  from  "liquids," 
and  yet  one  of  them  is  included  in  the  other.     All  that  is 
necessary  to  make  the  premises  negative  is  to  have  the  major 
and  minor  terms  excluded  from  the  same  middle  term,  and 
this  leaves  the  connection  between  the  major  and  minor  terms 
entirely  indeterminate.     We  can,  therefore,  draw  no  conclusion 
at  all,  either  affirmative  or  negative,  because  there  is  nothing 
said  or  denied  in  the  premises  to  imply  one  or  the  other. 

There  is  an  apparent  exception  to  Rule  6  in  such  cases  as 
the  following,  which  require  analysis. 

That  which  is  not  wise  cannot  be  useful. 

Intemperance  is  not  wise. 

.  * .  Intemperance  cannot  be  useful. 

Here  it  seems  very  apparent  that  we  have  negative  premises, 
and  yet  the  mind  does  not  wince  at  the  conclusion,  nor  feel 
that  anything  wrong  has  occurred  in  drawing  it.  The  reason- 
ing resists  all  attempts  to  invalidate  it.  But  the  reason  for 
this  is,  that  although  the  propositions  seem  to  be  EEE,  they 
are  in  reality  EAE  of  the  first  figure,  and  therefore  valid. 
The  first  proposition  is  made  negative  by  the  "cannot,"  but  in 


180  ELEMENTS  OF  LOGIC 

the  second  the  nature  of  the  middle  term  is  such  that  in  order 
to  have  it  the  same  in  both  premises  the  predicate  of  the  mi- 
nor premise  must  be  negative,  and  the  "  not "  in  its  real  im- 
port must  go  with  the  "  wise."  The  logical  subject  in  the 
major  premise  is  the  whole  thought  expressed  by  the  middle 
term,  which  is  "  the  not-wise  acts,"  and  in  the  minor  premise 
this  conception  can  be  brought  out  only  by  construing  the 
predicate  in  the  same  way,  and  so  regarding  it  as  equivalent 
to  "  that  which  is  not  wise,"  or  "  one  of  those  things  which  is 
not  wise,"  and  therefore  "  a  not-wise  act,"  and  we  find  that  the 
proposition  becomes  A  with  a  negative  conception  for  the 
predicate.  That  is,  we  simply  obvert  the  minor  premise,  so 
that  it  becomes  in  reality  affirmative.  "Whatever  we  may  think 
of  the  form  of  the  proposition,  therefore,  it  becomes  mate- 
rially an  affirmative  one,  or  is  thought  as  such,  and  the  rea- 
soning is  determined  accordingly. 

The  fact  calls  attention  to  the  double  use  of  the  negative 
particle  in  propositions  which  must  be  reckoned  with  in  deal- 
ing with  them.  It  may  be  attached  to  the  copula,  denying 
the  connection  between  subject  and  predicate,  and  making  the 
quality  of  the  judgment  negative,  or  it  may  be  attached  to  the 
predicate,  making  the  quality  of  the  judgment  affirmative  and 
the  predicate  negative.  This  is  but  a  case  of  obversion,  but  it 
shows  that  we  may  often  mistake  the  mere  form  of  a  proposi- 
tion for  its  matter.  We  must,  therefore,  be  on  our  guard 
against  mistaking  what  is  really  a  negative  attribute  for  a  neg- 
ative assertion. 

Rule  7  is  proved  and  illustrated  by  Figs.  19  and  21.  A  viola- 
tion of  it  would  appear  in  the  attempt  to  draw  the  affirmative 
conclusiou,  "Wood  is  iron,"  because  in  the  premises  "  wood  " 
is  excluded  from  "metals,"  and  so  must  be  excluded  from 
whatever  is  contained  in  that  class. 

Rules  8  and  9  can  be  best  tried  and  illustrated  when  we 
come  to  consider  the  Moods  and  Figures  of  the  syllogism  in 
the  next  chapter.  The  circles  would  take  up  more  space  than 
is  necessary,  while  practically  few  fallacies  are  incident  to  the 
8th  Rule. 


CHAPTER  XH. 

MOODS  AND  FIGURES  OP  THE  SYLLOGISM 

1st.  Moods. — The  Mood  of  a  syllogism  is  that  characteris- 
tic of  it  which  is  determined  solely  by  the  quantity  and  quality 
of  its  propositions.  The  Mood  can  never  be  separated  from 
the  Figure,  but  it  is  not  determined  by  the  same  properties. 
It  is  necessary  to  consider  both  of  them  in  order  to  ascertain 
the  valid  and  invalid  forms  of  reasoning,  but  this  result  can 
be  most  easily  and  most  briefly  attained  by  first  considering 
the  Moods.  Every  syllogism,  as  we  have  seen,  must  contain 
three  propositions,  and  only  three.  But  there  are  four  forms 
of  propositions,  namely,  A,  E,  I,  and  O,  from  which  to  choose, 
or  which  may  be  combined  in  various  ways  to  produce  the  re- 
quisite number  for  a  syllogism.  Thus  it  is  possible  to  take  all 
three  propositions  in  A.  In  such  a  case  the  major  and  minor 
premises,  and  the  conclusion,  would  each  and  all  be  proposi- 
tions in  A.  Or,  we  might  have  one  in  A,  one  in  E,  and  one  in 
O.  It  is  conceivable  that  all  three  propositions  should  be 
E,  or  I,  or  O.  We  happen  to  know,  however,  from  Rules  6 
and  8,  that  such  forms  are  not  valid  ;  but  apart  from  those 
rules  we  could  not  so  decide  the  matter.  Hence  we  must  rep- 
resent  at  present  all  the  conceivable  moods,  and  test  them  af- 
terward. Apart  from  these  rules,  therefore,  any  combination 
of  three  propositions  is  possible,  as  representing  the  conceiva- 
ble ways  in  which  a  syllogism  might  be  formed.  If  the  three 
propositions  were  in  A,  the  syllogism  for  AAA  would  repre- 
sent a  mood  in  which  the  major  premise  was  A,  the  minor 
premise  A,  and  the  conclusion  A.  This  order  represents  the 
order  of  the  premises  and  the  conclusion.  The  mood  AEO 
would  therefore  mean  that  the  major  premise  was  A,  the  mi- 
nor E,  and  the  conclusion  O.  Now  as  the  conceivable  moods 
may  represent  all  possible  combinations,  either  of  the  same 
kind  or  of  different  kinds  of   propositions,  AEO  being  one 


1S2  ELEMENTS  OF  LOGIC 

mood  an  EAO  being  another,  there  will  be  a  large  number 
of  possible  moods.  When  completed  we  find  the  possible 
number  to  be  (!4.     They  appear  as  follows  : 

AAA  AEA  AIA  AOA  EAA  EEA  EIA  EOA 

AAE  AEE  ATE  AOE  EAE  EEE  EIE  EOE 

AAI  AEI  All  AOI  EAI  EEI  EH  EOI 

AAO  AEO  AIO  AOO  EAO  EEO  EIO  EOO 


IAA 

IEA 

HA 

IOA 

OAA 

OEA 

OIA 

OOA 

IAE 

IEE 

HE 

IOE 

OAE 

OEE 

OLE 

OOE 

IAI 

IEI 

m 

IOI 

OAI 

OKI 

on 

001 

IAO 

IEO 

no 

IOO 

OAO 

OEO 

010 

000 

But  these  are  not  all  valid.  Some  of  them  violate  one  rule, 
and  some  another.  For  example,  EEA  violates  Rule  6  ;  EAI 
violates  Rule  7  ;  IOA  violates  Rules  7  and  8.  By  thus  ap- 
plying the  rules  of  the  syllogism  to  each  one  we  find  that 
a  large  number  of  them  are  to  be  rejected  as  p8(  udo^mood8}  or 
false  moods,  because  no  conclusion  can  be  valid  in  them.  In 
this  way  52  are  rejected  because  they  violate  some  one  or  two 
of  the  Rules  6,  7,  8,  and  9.  Some  of  them  also  violate  Rule  5, 
but  this  law  is  not  applied  until  we  come  to  test  the  moods  in 
the  Figures  of  the  syllogism.  There  remain,  therefore,  12 
possibly  valid  moods  ;  we  say  possibly,  because  it  is  not  proved 
that  they  are  valid  when  they  are  not  found  to  conflict  with 
any  of  the  above  four  rules.  They  must  first  be  tested  in 
the  Figures,  and  it  will  then  be  found  that  some  of  them  are 
valid  and  some  are  invalid  in  one  or  more  of  the  Figures,  and 
one  mood,  IEO,  proves  to  be  invalid  in  all  of  them,  on  the 
ground  of  an  illicit  process  of  the  major  term.  The  follow- 
ing are  the  12  moods,  with  IEO  bracketed  for  the  reason  just 
specified  : 

AAA  EAE  IAI  OAO 

AAI  EAO  [IEO] 

AH  EIO 

AEE 

AEO 

AOO 


MOODS  AND  FIGURES  OF  THE  SYLLOGISM      1S3 

These  remain  to  be  tested  by  the  Figures,  and  as  there  are 
four  Figures  there  will  be  48  forms  which  have  yet  to  be  con- 
sidered, four  of  which  have  to  be  rejected  as  involving  IEO, 
and  leaving  44  cases  in  which  the  reasoning  might  be  valid. 
But  as  a  large  number  are  found  to  violate  Rule  5,  in  one  or 
more  of  the  Figures,  they  have  to  be  rejected.  What  remains 
will  appear  in  a  moment. 

2d.  Figures. — The  Figure  of  a  syllogism  is  that  character- 
istic of  it  which  is  determined  by  the  position  of  the  middle 
term.  As  there  are  two  propositions,  each  with  a  subject  and 
predicate,  in  the  premises  of  every  syllogism,  there  are  four 
possible  positions  for  the  middle  term.  It  maybe  the  subject 
of  both,  the  predicate  of  both,  the  subject  of  the  major,  and 
the  predicate  of  the  minor,  or  the  predicate  of  the  major  and 
subject  of  the  minor  premise.  These  positions  are  repre- 
sented in  the  following  diagrams  : 


1st  Fig. 

2d  Fig. 

3d  Fig. 

4th  Fig. 

M  =  P 

P  =  M 

M  =  P 

P  =  M 

S  =  M 

S  =  M 

M  =  S 

M  =  S 

S  =  P 

S  =  P 

S  =  P 

S  =P 

Each  of  the  12  moods  can  be  tested  in  the  four  Figures, 
making,  as  already  said,  48  possible  forms  in  all,  which  might 
be  valid,  were  not  some  of  them  violations  of  Eule  5  in  one 
or  more  of  the  Figures.  Thus  in  mood  AAA  we  may  have 
four  different  positions  of  the  middle  term,  which  is  repre- 
sented in  the  following  diagrams,  showing  those  in  which  it 
is  valid  and  those  in  which  it  is  invalid  : 

1st  Fig.  2d  Fig.  3d  Fig.  4th  Fig. 

A@  =  P        A(p)  =  M        A@  =  P        A@  =  M 
A(§)=M       A®  =  M        A@  =  S        A@  =  S 
A©=P        A@  =  P        A(§)  =  P        A®  =  P 

Valid.  Invalid.  Invalid.  Invalid. 

In  the  second  Figure,  as  shown,  AAA  gives  rise  to  the  fal- 
lacy of  undistributed  or  Illicit  Middle  ;  in  the  third  Figure, 
to  the  fallacy  of  Illicit  Minor  ;  and  also  Illicit  Minor  in  the 


184  h'Lh'Mh'.XTS   OF   LOGIC 

fourth  Figure.    It  is  therefore  valid  formally  only  in  the  fourth 
Figure.     Take  another  example,  the  mood  AEE  : 

1st  Fig.  2d  Pig.  3d  Fig.  4th  Fig. 

A(M)  =  P        A(£)  =  M        A@=P        A®=M 

E®x(M)   E@x(g)   E@x®   E@X® 
E®X®   E®X®   E®X®   E®X® 

Invalid.  Valid.  Invalid.  Valid. 

We  find  AEE  valid  only  in  the  second  and  fourth  Figures. 
In  the  first  and  third  Figures  it  gives  rise  to  the  fallacy  of  Il- 
licit Major.  By  testing  each  mood  in  this  way  we  reject  a 
large  number  of  them  as  invalid  in  some  Figures,  although 
valid  in  others.  Only  one  is  not  valid  in  any  of  them,  and 
this,  as  remarked,  is  IEO.  When  all  the  valid  moods,  there- 
fore, are  selected  out  of  the  44,  we  find  there  are  24  of  them, 
which  are  given  below.  Of  this  number  5  are  called  Weak- 
ened Conclusions,  because,  although  valid,  they  are  of  no  prac- 
tical use,  because  &  particular  conclusion  is  drawn  when  a  uni- 
versal one  might  as  well  have  been  drawn.  They  are  placed 
in  brackets. 


1st  Fig. 

2d  Fig. 

3d  Fig. 

4th  Fig. 

AAA 

AEE 

AAI 

AAI 

AH 

AOO 

AH 

AEE 

EAE 

EAE 

IAI 

IAI 

[EAO] 

[EAO] 

EAO 

EAO 

EIO 

EIO 

EIO 

EIO 

[AAIJ 

[AEO] 

OAO 

[AEO] 

Several  other  lists  with  a  slightly  different  order  might  be 
given,  but  the  present  one  is  probably  convenient  enough 
for  the  memory.  The  arrangement  has  been  made  as  nearly 
as  possible  to  indicate  in  the  same  line,  and  with  as  little  space, 
those  moods  which  are  valid  in  more  than  one  Figure,  and  at 
the  same  time  to  collect  allied  moods  together,  so  far  as  per- 
missible. But  since  it  may  appear  a  difficult  task  to  memor- 
ize the  whole  24  forms,  and  to  locate  them  in  their  right  Fig- 
ures. I  have  adopted  a  mnemonic  system  which  may  be  con- 


MOODS  AND  FIGURES  OF  THE  SYLLOGISM      185 

venient  for  many  students,  although  I  do  not  find  it  necessary 
to  use  it  for  myself.  The  meaning  of  the  system  can  be  easily 
explained.  The  vowels  indicate  the  Moods;  the  consonants 
B,  C,  D,  and  M  represent,  respectively,  the  first,  second,  third, 
and  fourth  Figures,  M  being  chosen  instead  of  F,  solely  for 
the  sake  of  euphony.  L  and  N  are  simply  connecting  conso- 
nants where  we  cannot  use  B,  C,  D,  or  M.  There  are  only 
eleven  words  in  aU  in  the  system,  corresponding  to  the 
eleven  valid  moods.  Each  word  indicates  the  mood  and  Fig- 
ures in  which  a  given  form  is  valid.     They  are  as  follows :  * 


Balana 

Caleme 

Dilami 

Badami 

Calemo 

Dolano 

Badini 

Calono 

Becane 

Becadom 

Becidom 

The  shortest  possible  list  of  the  valid  moods  is  the  following, 
and  it  will  not  appear  so  formidable  to  the  memory  as  the 
previous  instances.  It  ignores  the  weakened  conclusions  be- 
cause it  is  apparent  that  I  can  be  drawn  wherever  A  is  drawn, 
an  O  wherever  E  is  drawn.  The  numbers  following  each  mood 
signify  the  Figures  in  which  it  is  valid.  These  moods  can 
probably  be  best  remembered  without  resort  to  a  mnemonic 
system. 

AAA,  1.         AEE,  2.  4.       EAE,  1.  2.  IAI,  3.  4. 

AH,  1.  3.       AOO,  2.  EIO,  1.  2.  3.  4.       OAO,  3. 

In  examining  the  list  of  valid  moods  and  Figures,  we  ob- 
serve that  the  first  Figure  is  the  only  one  which  can  have  A  for 
a  conclusion,  and  is  the  only  Figure  in  which  all  of  the  four 
propositions  A,  E,  I,  O  can  be  proved.  In  the  second  Figure 
we  observe  only  negative  conclusions.     No  affirmative  conclu- 

*  This  is  not  intended  to  take  the  entire  place  of  the  usual  mnemonic 
verses  to  be  considered  under  the  reduction  of  the  Moods  and  Figures, 
but  only  to  aid  in  the  detection  of  those  which  are  valid  in  more  than  one 
Figure.  Those  who  adopt  the  common  system  need  not  adopt  the  one  I 
have  given. 


186  ELEMENTS  OF  LOGIO 

sion  is  possible  in  this  Figure  because  the  premises  would 
have  to  be  affirmative  in  order  to  give  it.  But  since  the  middle 
term  is  the  predicate  in  both  premises  of  this  Figure,  it  is  un- 
distributed whenever  the  proposition  is  affirmative,  and  hence 
an  inference  would  be  a  case  of  illicit  middle.  The  third 
Figure  gives  only  particular  conclusions.  If  the  major  prem- 
ise be  negative,  a  universal  conclusion  would  be  an  illicit 
minor  ;  if  the  minor  premise  be  negative,  any  conclusion  would 
be  an  illicit  major.  Hence  only  particular  conclusions  can  be 
drawn,  and  that  only  when  both  premises  are  affirmative,  or 
the  major  premise  negative,  except  in  the  Mood  OAO.  The 
fourth  Figure  is  peculiar  in  being  easily  reducible  into  the 
first  Figure,  and  hence  it  is  of  little  practical  use.  "We  have 
only  to  change  the  position  of  the  premises,  or  m  utaie  them, 
as  it  is  called,  in  order  to  convert  the  fourth  Figure  into  the 
first.  It  requires  mention,  however,  because  certain  moods 
are  valid  in  it  which  are  not  valid  in  the  first. 

In  regard  to  the  practical  importance  and  usefulness  of  the 
Figures  there  is  some  difference  between  them.  "We  have  al- 
ready remarked  that  the  first  Figure  is  the  only  one  to  give  con- 
clusions in  all  four  propositions,  A,  E,  I,  and  O  ;  the  first  will 
give  them  in  E  and  O ;  the  third  in  I  and  O  ;  and  the  fourth 
in  E,  I,  and  O.  The  first  Figure,  therefore,  is  the  only  oue  to 
give  universal  affirmatives,  or  A.  This  fact  makes  it  the  most 
important  and  useful  of  the  forms  of  reasoning,  because  the 
demands  of  human  belief  and  conduct  require  universal  truths 
of  some  kind  which  do  not  give  excej)tions  such  a  place  and 
influence  as  to  impair  the  value  of  general  principles.  Thus  if 
the  needs  of  society  require  that  "All  men  should  do  right," 
and  this  proposition  was  not  accepted  on  its  bare  statement, 
we  should  require  to  prove  it,  and  upon  premises  of  which  it 
was  a  part  or  an  equal.  But  if  we  could  prove  from  those 
premises  only  that  "Some  men  should  do  right,"  the  truth 
would  be  so  indefinite  that  there  would  be  no  telling  whether 
the  obligation  was  incumbent  upon  a  sufficient  number  of 
men  to  make  the  principle  worth  asserting.  If  the  "  some  " 
means  two  or  three,  or  a  hundred,  the  immunities  which  the 


MOODS  AND  FIGURES  OF  THE  SYLLOGISM      187 

remaining  majority  may  have,  so  far  as  premises  and  con- 
clusion are  concerned,  make  the  principle  perfectly  ineffect- 
ive, especially  as  the  particularity  and  indefiniteness  of  the 
conclusion  is  such  as  to  make  it,  practically,  a  universal  nega- 
tive ;  that  is,  if  we  only  prove  that  "  Some  men  should  do  right," 
and  are  not  able  to  specify  what  "  some  "  or  portion  of  them 
are  under  that  obligation,  the  indefiniteness  of  the  assertion 
prevents  us  from  asserting  the  obligation  of  anybody  in  par- 
ticular. But  if  we  can  prove  the  universal,  the  conclusion  ap- 
plies definitely  to  everybody.  Hence  the  first  Figure  has  an 
importance  corresponding  to  its  efficiency  for  establishing 
such  conclusions. 

There  is  another  point  of  interest  in  this  Figure  to  which 
Keynes  calls  attention.  It  is  that  only  in  it  have  we  both  the 
subject  of  the  conclusion  as  the  subject  in  the  premises,  and 
the  predicate  of  the  conclusion  as  the  predicate  in  the  prem- 
ises. This  makes  it  unnecessary  for  the  mind  to  change  its 
conception  from  one  relation  to  the  other  in  coming  to  its 
conclusion  ;  it  can  still  think  of  its  terms  in  the  conclusion  as 
having  the  same  functions  and  logical  relations  as  in  the  prem- 
ises. It,  therefore,  accounts  for  the  fact  that  reasoning  so 
generally  seems  more  natural  in  the  first  Figure  than  in  any 
other.  We  resort  to  it  for  greater  clearness  and  effective- 
ness. 

The  first  Figure  is  especially  adapted  to  disproof,  or  the  proof 
of  universal  negatives,  or  propositions  in  E.  It  is  so  adapted, 
partly  because  one  of  the  premises  must  be  negative,  and  partly 
because  of  the  kind  of  comparison  which  can  be  established 
by  it  between  subjects  and  predicate.  It  is  clear  that  if  two 
things  cannot  agree  in  their  predicate  they  cannot  agree  with 
each  other.  Hence  to  disprove  an  assertion  we  have  only  to 
prove  that  one  instance,  or  more,  included  in  the  general  state- 
ment does  not  agree  with  the  predicate  and  we  have  the  con- 
tradictory established.  Thus,  if  it  be  asserted  that  "All 
forms  of  government  are  beneficial,"  we  may  disprove  this  by 
showing  that  "An  arbitrary  despotism  is  not  beneficial," 
which  would  involve  in  a  comparison  of  the  two  statements  a 


188  ELEMENTS  OF  LOGIC 

syllogism  of  the  second  Figure,  whose  conclusions  would  be, 
"  Arbitrary  despotisms  are  not  governments,"  which  is  the  op- 
posite of  what  is  implied  in  the  nature  of  the  case,  and  hence 
the  opponent  will  have  the  option  of  giving  up  the  universal- 
ity of  his  assertion  that  "  All  governments  are  beneficial,"  or 
of  maintaining  that  despotisms  are  not  forms  of  government. 

This  seems,  however,  like  a  mere  proof  of  an  excejition,  and 
not  of  the  total  falsity  of  the  universal.  A  better  instance  of 
the  case  is  the  following.  Suppose  it  be  asserted  that  "  Strikes 
are  justifiable."  The  most  complete  disproof  of  this  would 
be  the  truth  of  the  proposition  that  "No  strikes  are  justifi- 
able." Now,  if  we  can  assert  that  "  All  interferences  with  the 
rights  of  property  and  capital  are  not  justifiable,"  and  can  have 
it  taken  for  granted  that  strikes  are  such  interferences,  we  have 
proved  the  direct  opposite  of  the  original.  The  syllogism 
would  stand  thus  with  the  original  proposition  as  the  minor 
premise,  instead  of  the  major,  as  in  the  first  case. 

Interferences  with  the  rights  of  property  and  capital  are  not 

justifiable. 
Strikes  are  justifiable. 
.*.  Strikes  are  not  justifiable. 

Now  as  the  minor  premise  is  asserted  by  one  person  and 
the  major  by  another,  with  the  supposition  that  it  will  be  ad- 
mitted by  the  first  person,  the  conclusion  intended  to  be  en- 
forced is,  "  Strikes  are  not  interferences  with  the  rights  of 
property  and  capital,"  while  it  is  assumed  that  it  is  clear  they 
are  such  interferences.  Hence  if  they  are  such,  and  yet  the 
major  premise  is  admitted,  it  will  follow  that  "All  strikes  are 
not  justifiable,"  the  direct  opposite  of  the  original  proposition 
asserted.  The  second  Figure  is,  therefore,  well  adapted  to  the 
method  of  disproof. 

The  third  Figure  is  adapted  to  the  proof  of  exceptions  to 
universals,  or  disproves  universals  by  their  contradictories,  as 
the  first  Figure  disproves  them  by  their  opposites.  Or,  when 
it  does  not  prove  exceptions,  it  proves  certain  particular  truths 
which  will  be  of  considerable  value  against  the  assertion  of 


MOODS  AND  FIGURES  OF  THE  SYLLOGISM       1S9 

this  universal  contradictory.  As  an  illustration  of  this  latter 
fact,  as  well  as  the  former,  take  the  assertion  that  "  No  philos- 
ophers are  wise."  Now,  if  we  wish  to  disprove  this,  and  at  the 
same  time  prove  that  "  Some  philosophers  are  wise,"  we  have 
only  to  prove  the  following  : 

Plato,  etc.,  were  wise. 

Plato,  etc.,  were  philosophers. 

v  Some  philosophers  were  wise. 

Here  we  both  prove  that  "  Some  philosophers  are  wise,"  and 
disprove  the  universal  negative  which  had  been  asserted 
against  this  possible  fact.  A  particular  exception  is  proved. 
Therein  lies  the  value  of  the  third  Figure,  when  it  is  found 
difficult  or  impossible  to  prove  the  universal  opposite  of  a 
given  assertion. 

The  fourth  Figure  is  not  regarded  by  logicians  as  having  any 
great  practical  value.  It  is  not  so  frequently  used  as  the 
others,  and  can  so  often  be  changed  into  the  first  Figure  by 
mutating  the  premises,  that  its  practical  importance  does  not 
require  special  notice. 


CHAPTER  XIQ. 

REDUCTION   OF   MOODS   AND   FIGURES 

1st.  Direct  Reduction. — The  number  of  valid  Moods  and 
Figures  is  so  great,  and  the  apparently  irregular  character  of 
them  so  perplexing  that  the  mind  often  finds  it  difficult  to  re- 
member them  correctly.  The  old  logicians,  therefore,  in- 
vented a  mnemonic  device  in  aid  of  the  memory,  and  which 
at  the  same  time  would  contain  some  indications  of  the  vari- 
ous processes  required  for  what  is  called  their  reduction. 
The  mnemonic  verses  consist  of  barbarous  Latin  terms  added 
to  a  few  genuine  ones,  indicating  all  the  Moods  and  Figures 
that  are  valid,  except  the  instances  of  weakened  conclusion. 
The  verses  are  given  below,  with  the  indications  of  the  accent 
to  be  employed  in  reading  them  according  to  the  laws  for 
scanning  Latin  poetry  : 

Barbara,  Celaront,  DariT3  Fcrioquc  prioris  : 
Cosare,  Camestres,  Fcstfnu,  BarokS,  secundse : 
Tertia,  DaraptI,  Disarms,  Datlsi,  Felapton. 
Bokardo,  Feiison,  habet  :  Quarta  insiiper  addit 
BramantTp,  Camenes,  Dlmarls,  Fesapo,  Fresison. 

The  first  line  indicates  the  moods  of  the  first  Figure,  the 
second  the  moods  of  the  second  Figure  ;  the  third,  and  the 
first  two  words  of  the  fourth  line,  the  moods  of  the  third  Fig- 
ure, and  the  last  line  those  of  the  fourth  Figure.  The  first 
Figure  is  called  by  logicians  the  perfect,  and  the  remaining 
threo  the  imperfect  Figures,  because  they  are  to  be  reduced 
by  certain  processes  to  the  first.  The  letters  indicating  how 
this  is  to  be  done  require  to  be  explained. 

The  capital  letters  B,  C,  D,  F,  in  the  last  four  lines,  indicate 


REDUCTION  OF  MOODS  AND  FIGURES  191 

the  mood  in  the  first  Figure  to  which  the  mood  in  the  other 
Figures  beginning  with  that  letter  is  to  be  reduced.  Thus 
Camestres  is  to  be  reduced  to  Celarent,  etc. 

The  vowels  a,  e,  i,  o  indicate  the  mood  of  the  syllogism. 
Thus  in  Ferison  the  mood  is  EIO  etc. 

p  indicates  that  the  preceding  proposition  is  to  be  converted 
per  accidens  or  by  limitation. 

s  at  the  end  of  a  word  indicates  that  the  conclusion  of  the 
new  syllogism  has  to  be  converted  by  simple  conversion  in 
order  to  obtain  the  given  conclusion  :  in  the  middle  of  a  word 
it  denotes  that  the  preceding  proposition  is  to  be  converted 
simply  in  the  process  of  reduction.  This  difference  between 
the  use  of  s  at  the  end,  and  its  use  in  the  middle  of  a  word  is 
remarked  by  Keynes.  But  I  am  inclined  to  see  little  impor- 
tance in  the  distinction,  because  s  in  both  cases  denotes  that 
the  preceding  proposition  is  to  be  converted  simply,  so  that 
the  converted  conclusion  of  the  reduced  mood  becomes  the 
proper  conclusion  of  the  corresponding  "perfect"  Figure  and 
Mood.  Thus  in  Camestres  we  are  to  reduce  the  mood,  as  al- 
ready said,  to  Celarent,  or  AEE  of  the  second  Figure  to  E 
AE  of  the  first.  The  m  denoting  that  the  premises  are  to  be 
transposed,  we  have  to  convert  the  minor  premise  and  make 
it  the  major  premise  of  the  new  syllogism,  and  make  the  ma- 
jor of  Camestres  the  minor  of  the  new.  Then,  if  we  convert 
the  conclusion  of  Camestres  by  simple  conversion,  we  shall 
have  the  proper  conclusion  of  the  new. 

Camestres.  Celarent. 

AisC         |  (       CisnotB 

B  is  not  C  >•  Reduced  to  -l       A  is  C 
.  • .  B  is  not  A )  ' .  • .  A  is  not  B. 

Keynes  also  makes  the  same  distinction  between  the  use  of 
p  at  the  end,  and  its  use  in  the  middle  of  a  word.  But  as  in 
the  case  of  s,  I  do  not  find  it  important.  In  all  cases  it  de- 
notes that  the  preceding  proposition  is  to  be  converted  by  lim- 
itation, and  so  to  appear  in  the  new  syllogism.  Thus  in  Felap- 
ton  A  is  to  be  converted  per  accidens  or  by  limitation.     In 


192  ELEMENTS  OF  LOGIC 

Bramantip,  however,  the  subalternans  of  the  converse  is  taken 
after  the  conversion. 

m  denotes  that  the  premises  of  the  "  imperfect "  moods  are 
to  be  transposed  or  mutated  in  order  to  form  the  premises  of 
the  new  and  "perfect"  mood.  Thus  in  Camenes  A  and  E  must 
be  transposed  so  as  to  become  E  and  A  of  Celarent. 

k  signifies  that  the  mood  is  to  be  reduced  indirectly  ;  and 
the  position  of  the  letter  is,  as  affirmed  by  Keynes,  to  indicate 
that  in  the  process  of  indirect  reduction  the  first  step  is  to 
omit  the  premise  preceding  it ;  that  is,  we  take  instead  of  it 
the  contradictory  of  the  conclusion  as  our  premise.  Baroko 
and  Bokardo  are  the  moods  to  which  indirect  reduction  is 
usually  applied,  although  the  j>rocess  can  be  applied  to  others. 
The  present  mnemonic  lines,  however,  do  not  designate  how 
this  can  be  done  in  the  case  of  the  other  moods.  They  can  be 
more  easily  reduced  directly,  and  it  is  superfluous  to  apply  the 
indirect  process. 

One  or  two  more  examples  of  direct  reduction  will  suffice 
for  illustration,  and  we  can  pass  to  consider  the  indirect  pro- 
cess. Take  the  cases  of  Festino  and  Bramantip,  which  are  to 
be  reduced  respectively  to  Ferio  and  Barbara.  If  we  perform 
the  processes  as  designated  by  the  several  letters  they  will 
stand  as  follows : 

Festino.  Ferio. 

A  is  not  C  \  T  C  is  not  A 

Some  B  is  C  >  Reduced  to  a  Some  B  is  C 

.  • .  Some  B  is  not  A )  ' .  • .  Some  B  is  not  A. 

Bramantip.  Barbara. 

A  is  C  \  ( C  is  B 

C  is  B  V  Reduced  to  A  A  is  C 

Some  B  is  A )  ' .  ■ .  Some  A  is  B, 

or  All  A  is  B.       Subalternans. 

In  the  case  of  Bramantip  there  is  apparent  a  defect  in  the 
mnemonic  verse,  as  the  term  ought  to  indicate  the  process  of 
changing  the  converted  I  into  its  subalternans  A.  One  or 
two  concrete  exanrples  will  be  helpful  to  a  better  understand- 


REDUCTION  OF  MOODS  AND  FIGURES  193 

ing  of  the  process.  We  may  take  Gesare  and  Disamis,  to  be 
reduced  to  Gelarent  and  Darii.  Following  the  directions,  we 
have  in  Gesare  and  Gelarent  the  two  succeeding  syllogisms  : 

Cesare.  Celarent. 

Men  are  not  quadrupeds.  \  C  Quadrupeds  are  not  men. 

Horses  are  quadrupeds.      V  Reduced  to  -j  Horses  are  quadrupeds. 
.  ■ .  Horses  are  not  men.     )  (  .  • .  Horses  are  not  men. 

Disamis.  Darii. 

Some  men  are  negroes.  }  (  All  men  are  vertebrates. 

All  men  are  vertebrates.  >  Reduced  to  •<  Some  negroes  are  men. 

.  • .  Some  vertebrates  are  negroes.  )  (  .  ■ .  Some  negroes  are  vertebrates. 

The  process  can  easily  be  applied  to  the  other  moods  with- 
out further  illustration.  It  will  be  good  practice  for  the  stu- 
dent to  choose  his  own  examples. 

2d.  Indirect  Reduction. — Some  difficulty  attends  the  re- 
duction of  Baroko  and  Bokardo,  because  there  are  no  moods 
in  the  first  Figure  representing  AOO  and  OAO,  the  former 
giving  an  illicit  major  and  the  latter  an  illicit  middle.  Besides, 
no  conversion  of  O  is  possible,  and  hence  if  the  reduction  of 
these  two  moods  be  possible  it  must  be  in  some  indirect  way. 

There  are  two  ways  in  which  it  is  done,  and  there  is  a  dif- 
ference of  opinion  as  to  whether  the  first  form  is  indirect  or 
not.  Jevons  speaks  of  it  as  indirect ;  Keynes  as  direct.  But 
inasmuch  as  the  major  premise  in  both  moods  is  contra  verted, 
and  the  minor  premise  of  Baroko  obverted,  it  will  hardly  be 
amiss  to  follow  Jevons's  opinion,  because  neither  of  these  pro- 
cesses are  provided  for  in  the  mnemonic  system  we  have 
adopted.  If  the  matter  of  reduction  were  of  great  practical 
importance,  it  might  be  incumbent  upon  us  to  discuss  this 
question.  But  the  chief  matter  of  interest  to  the  scientific 
student  of  Logic  is  the  fact  and  the  mode  of  reduction,  and 
we  content  ourselves  with  stating  them. 

We  take  up  the  first  method  of  treating  Baroko  and  Bokar- 
do, illustrating  them  as  before : 

Baroko.  Fcrio. 

A  is  C  j)  (  All  not-C  is  not  A 

Some  B  is  not  C      ^  Reduced  to  <  Some  B  is  not-C 
.*.  Some  B  is  not  A  )  (  .-.  Some  B  is  not  A. 

13 


194  ELEMENTS  OF  LOGIC 

It  should  first  be  remarked  in  tbis  case  that  the  mnemonic 
Baroko  is  named  so  and  introduced  with  the  capital  B  to  suit 
its  indirect  reduction  to  Barbara.  But  in  reducing  it  to  Ferio, 
it  should  be  called  Faroko.  The  obversion  of  the  minor  prem- 
ise is  accomplished  as  usual  by  connecting  the  negative  par- 
ticle with  the  predicate,  making  the  proposition  affirmative. 
We  give  a  concrete  illustration  of  the  same  : 

Baroko.  Ferio. 

Metals  are  elements.  \  (  All  not-elements  are  not  metals. 

Some  solids  are  not  elements.  V  -j  Some  solids  are  not-elements. 

.  • .  Some  solids  are  not  metals.  )  (  •  '  •  Some  solids  are  not  metals. 

The  reduction  of  Bokardo  by  the  same  method  is  as  follows, 
except  that  we  have  to  transpose  the  premises : 

Bokardo.  Darii. 

Some  A  is  not  C        )  ( All  A  is  B 

All  A  is  B  V  Reduced  to  a  Some  not-C  is  A 

.-.  Some  B  is  not  C  )  L\  Some  not-C  is  B. 

A  similar  remark  should  be  made  about  Bokardo  as  was 
made  about  Baroko.  To  suit  the  reduction  we  have  given,  it 
should  be  called  Dokardo.  We  also  give  a  concrete  illustra- 
tion of  the  process  : 

Bokardo.  Darii. 

Some  trees  are  not  oaks.  )  Tt  A       A  (  -^  trees  are  organisms- 

All  trees  are  organisms.  ft)  Some  not-oaks  are  trees. 

.  '.  Some  organisms  are  not  oaks.  )  (  •  '  ■  Some  not-oaks  are  organisms. 

The  conclusion  in  the  form  to  which  Bokardo  is  reduced  is 
also  contraverted,  as  can  be  observed  hj  the  student. 

The  second  method  of  reducing  the  same  moods  is  un- 
doubtedly indirect,  as,  instead  of  depending  upon  the  forms 
of  conversion,  obversion,  or  contraversion,  it  resorts  to  the 
square  of  opposition  for  a  contradictory  of  the  minor  premise 
in  one  case,  and  of  the  conclusion  in  the  other  for  a  premise, 
as  the  illustrations  below  will  distinctly  show  : 

Baroko.  Barbara. 

AisC  ]  (       AisC 

Some  B  is  not  C  V  Reduced  to  K        B  is  A 
Some  B  is  not  A )  ' .  • .  B  is  C. 


REDUCTION  OF  MOODS  AND  FIGURES  195 

In  this  process  the  major  premise  remains  the  same.  But 
instead  of  taking  any  form  of  the  minor  premise,  converted  or 
otherwise,  the  contradictory  of  the  conclusion  is  assumed.  In 
Baroko  the  conclusion  is  O ;  its  contradictory  is  A.  This  is 
done  on  the  supposition  that  if  O,  "  Some  B  is  not  A,"  is  not 
true,  then  "  B  is  A  "  is  true,  and  if  this  be  assumed  hypo- 
thetically  in  the  premises  we  shall  have  a  syllogism  in  Bar- 
bara with  a  conclusion  which  is  the  contradictory  of  the 
omitted  minor  premise,  "  Some  B  is  not  C."  This  contradic- 
tory is  "  B  is  C,"  and  we  must  either  admit  one  of  our  prem- 
ises to  be  false  or  allow  that  our  original  conclusion  is  true. 
But  as  we  assume  that  our  original  premises  and  conclusion 
are  true,  the  impossibility  of  their  contradictories  being  true 
is  an  indirect  proof  of  them,  as  it  is  always  regarded  an  in- 
direct proof  of  a  proposition  to  prove  the  impossibility  of  its 
opposite  or  contradictory,  and  reduction  is  proving  one  mood 
and  Figure  by  another.  Bokardo  may  be  treated  in  a  manner 
similar  to  Baroko. 

Bokardo.  Barbara. 

Some  A  is  not  C )  (       All  B  is  C 

All  A  is  B  >•  Reduced  to  <       All  A  is  B 

.  • .  Some  B  is  not  C  )  ( .  • .  All  A  is  C. 

In  this  instance  the  major  premise  is  omitted,  and  in  its 
place  stands  hypothetically  the  contradiction  of  the  conclu- 
sion in  Bokardo  for  the  major  premise  of  Barbara.  We,  there- 
fore, have  a  conclusion  which  is  the  contradictory  of  "  Some  A 
is  not  C,"  the  major  premise  of  Bokardo.  If  this  premise  be 
assumed  to  be  true,  the  impossibility  of  the  truth  of  its  con- 
tradictory in  the  reduced  form  is  an  indirect  proof  of  it,  and 
we  have  the  indirect  reduction  as  before.  The  method  is 
sometimes  called  the  reduclio  per  impossibile,  or  the  proof  of 
a  thing  by  showing  its  contradictory  to  be  impossible. 

Little  or  no  practical  importance  can  be  attached  to  any 
forms  of  reduction,  except  when  we  find  it  necessary  to  test  a 
conclusion  which  does  not  seem  so  clear  to  us  in  one  Figure 
or  form  of  reasoning  as  in  another.     They  may  sometimes  be 


196  ELEMENTS  OF  LOGIC 

helpful  in  completing  certain  enthymemes  which  can  be 
formed  in  either  the  first  or  the  second  Figures,  and  be  valid 
in  the  first,  but  invalid  in  the  second.  But  as  reduction  does 
not  apply  to  invalid  moods  it  is  only  the  principle  of  it  that 
can  be  applicable  in  such  cases.  Hence  we  do  not  generally 
find  the  process  to  have  more  than  a  purely  scientific  interest 
and  importance. 


CHAPTEK  XIV. 

FORMS   OF   SYLLOGISTIC   REASONING 

The  syllogism,  as  it  has  already  been  explained,  appears  to 
have  a  very  simple  form,  and  to  consist  of  three  propositions 
arranged  in  a  particular  way.  While  this  is  the  regular  form 
of  reasoning,  and  expresses  the  real  nature  of  the  mental  pro- 
cess involved  in  all  instances  of  ratiocination,  arguments  are 
not  always  formulated  in  the  regular  way.  Some  propositions 
may  be  omitted  in  expression,  although  included  in  the 
thought  of  the  reasoner,  and  in  other  cases,  two  or  more  syl- 
logisms may  be  involved  in  the  argument.  These  two  condi- 
tions give  rise  to  two  divisions  of  the  syllogism,  besides  the 
simple  form  already  considered.  They  are  the  incomplete  and 
the  complex  forms  of  it.  We  shall  classify  them  before  dis- 
cussing them  in  detail.     The  following  is  an  outline  of  them : 

fp        I  ,     j  Simple  =  The  ordinary  form  already  discussed. 
P         "(  Complex  =  Prosyllogism  and  Episyllogism. 
(  1  st  order. 
Syllogisms  ,  gimple  =  Enthymeme  )  2d  order. 

Incomplete  )  {  od  order. 

1  (Complex -  |8££~ 

The  complete  forms  of  the  syllogism  are,  of  course,  an  ex- 
plicit statement  of  all  that  is  involved  in  the  mental  process. 
They  very  seldom  appear  in  ordinary  discourse,  and  then  only 
to  give  it  more  cogency.  The  usual  mode  of  stating  an  argu- 
ment is  either  to  state  the  facts  and  to  allow  the  inference  to 
be  drawn  by  others,  or  to  abbreviate  the  process  by  assuming 
the  most  apparent  of  the  premises.  But  when  we  wish  to  in- 
dicate with  perfect  clearness  all  that  the  mind  takes  into  ac- 
count we  formulate  our  thoughts  into  complete  syllogi 
The  first  of  these  is  the  simple  syllogism  of  three  propositions, 


198  ELEMENTS  OF  LOGIC 

and  which  has  already  been  discussed.     We  then  take  up  the 
second  to  consider  it  briefly. 

1.  Prosyllogism  and  Episyllogism. — This  consists  of  two 
syllogisms,  the  conclusion  of  one,  the  prosyllogism,  being  a 
premise  in  the  other,  the  episyllogism.  The  following  are 
illustrations : 

A  is  B  Men  are  vertebrates. 

C  is  A  Europeans  are  men. 

.  • .  C  is  B  .  • .  Europeans  are  vertebrates. 

D  is  C  Italians  are  Europeans. 

.  • .  D  is  B  .  * .  Italians  are  vertebrates. 

In  the  formal  illustration  here  given  the  jirosyllogism  ends 
with  the  conclusion  "  C  is  B,"  and  the  episyllogism  begins 
with  the  same  proposition  as  a  premise.  In  the  material  in- 
stance the  proposition,  "Europeans  are  vertebrates,"  is  the 
conclusion  of  the  prosyllogism  and  a  premise  of  the  episyllo- 
gism. 

2.  Enthymeme. — An  enthymeme  is  an  incomplete  syllogism 
in  which  either  one  of  the  premises,  or  the  conclusion,  may  be 
omitted.  If  the  major  premise  be  the  one  which  is  omitted 
the  enthymeme  is  said  to  be  of  the  first  order ;  if  the  minor 
premise  be  omitted,  it  is  of  the  second  order,  and  if  the  con- 
clusion be  omitted,  it  is  of  the  third  order.  The  signs  of  the 
enthymeme  are  such  words  as  indicate  a  reason  for  the  truth 
of  a  proposition.  They  are  for,  because,  since,  inasmuch  as, 
and  in  the  conclusion,  therefore,  consequently,  etc.  The  last 
two  words  denote  that  an  inference  is  drawn  from  some  pre- 
ceding fact  or  statement,  and  are  signs  of  an  enthymeme  only 
when  a  single  premise  is  given.  Those  which  give  a  reason 
for  a  truth  are  usually  necessary  when  the  syllogism  is  not 
complete,  or  when  the  conclusion  is  stated  first  and  requires 
to  have  its  dependence  upon  the  premises  indicated.  As  an 
illustration  of  an  enthymeme  we  have  the  proposition  "Atmos- 
pheric air  must  have  weight  because  it  is  a  material  substance." 
The  conclusion  in  this  example  is,  "Atmospheric  air  must  have 
weight."     If  we  were  stating  a  mere  fact,  it  would  not  require 


FORMS  OF  SYLLOGISTIC  REASONING  199 

proof,  but  we  often  desire  to  support  such  assertions  by  rea- 
sons that  will  show  their  truth  apart  from  the  acceptance  of 
them  on  authority,  and  so  we  give  a  reason  for  them.  In  this 
case  the  reason  for  the  truth  or  assertion  that  "  Atmospheric 
air  must  have  weight  "  is,  that  "  it  is  a  material  substance." 
In  this  reason  it  is  tacitly  assumed  that  "  all  material  sub- 
stances have  weight."  It  is  the  omission  of  this  premise  that 
makes  the  reasoning  an  enthymeme.  Whenever  such  proposi- 
tions are  taken  for  granted  it  is  because  they  either  require  no 
proof  or  they  are  sufficiently  evident  to  justify  their  omission, 
and  the  process  of  thought  or  reasoning  can  be  abbreviated  or 
economized.  It  is  the  usual  form  in  ordinary  speech.  When 
resolved  into  the  form  represented  by  their  proper  logical 
order  and  dependence,  the  propositions  take  the  order  laid 
down  for  the  common  syllogism.  The  above  instance  can 
easily  be  resolved  by  the  student  into  a  simple  syllogism. 

There  are  forms  of  the  enthymeme  in  which  the  signs  are 
not  expressed,  but  which  have  to  be  determined  by  the  evi- 
dent relation  of  the  thoughts  stated.  Every  sentence,  how- 
ever, which  is  definitely  introduced  by  a  particle  denoting  a 
reason  for  something  else  is  an  express  statement  of  reason- 
ing. But  such  forms  of  proof  become  stilted  and  inelegant, 
and  hence  discourse  may  sometimes  be  best  conducted  by  af- 
firming facts  or  truths  which  of  themselves  imply  the  de- 
pendence of  something  else  upon  them.  We  require  to  be  on 
the  alert  in  such  cases  in  order  either  to  detect  the  existence 
of  reasoning  at  all,  or  to  discover  the  form  in  which  it  is  im- 
plied. This  may  also  be  true  of  statements  involving  com- 
plete syllogisms.  An  interesting  instance  of  reasoning  in  the 
form  of  an  enthymeme  without  the  usual  signs  is  the  follow- 
ing: 

"  The  high  prices  caused  by  the  new  tariff  law  have  severely 
taxed  the  mind  of  the  Chronicle-Telegraph,  but  it  has  at  last 
evolved  this  curious  and  interesting  explanation  :  '  For  a  long 
time  past  everything,  or  nearly  everything,  entering  into  the 
daily  consumption  of  the  people  has  been  unusually  cheap.  But 
close  observex-s  of  the  trend  of  events  believe  we  are  approach- 


200  ELEMENTS  OF  LOGIC 

ing  an  era  of  higher  prices,  and  that  it  is  near  at  hand,  and 
that  it  is  to  be  looked  upon  as  a  natural  reaction  from  the  pe- 
riod of  constant  decline  in  the  value  of  merchandise  which 
has  lasted  for  several  years.'  "  The  reasoning  involved  in  this 
case  is  apparent  when  we  enunciate  that  the  present  high 
prices  are  a  reaction  from  previous  low  prices,  and  that  such 
reactions  are  natural,  and  not  caused  by  tariff  laws.  The  pas- 
sage beginning  with  the  words,  "For  a  long  time  past,"  is  in- 
tended to  express  a  fact,  from  which  it  follows  that  high  prices 
must  occur  without  reference  to  tariff  influences,  and  hence 
we  have  in  thought  a  process  of  reasoning  without  the  use  of 
any  of  the  signs.  The  premise  which  is  omitted  is,  that  "  All 
periods  of  extremely  low  prices  are  followed  by  a  natural  re- 
action." If  the  other  premise  be  formulated  we  should  all 
see  how  the  conclusion  would  follow,  which  is  intended  to  be 
the  contrary  of  the  first  proposition  in  the  passage.  Most  ar- 
guments in  ordinary  discourse  are  of  this  general  kind,  and 
the  student  would  do  well  to  formulate  them  for  the  sake  of 
discovering  where  the  reasoning  is. 

3.  Epicheirema. — An  epicheirema  is  a  syllogism  in  which 
one  or  both  of  the  premises  is  supported  by  a  reason  which 
implies  an  imperfectly  expressed  syllogism  :  in  other  words,  it 
is  a  syllogism  in  which  one  or  both  of  the  premises  is  an  en 
thymeme  either  of  the  first  or  of  the  second  order.  The  epi- 
cheirema may  be  single  or  double.  It  is  single  when  only  one 
of  the  premises  is  an  enthymeme,  and  double  when  both  are 
enthymemes.     The  following  is  an  illustration  of   the  single 

epicheirema : 

A  is  B,  for  it  is  P 

Cis  A 

.  • .  C  is  B. 

A  material  form  of  the  same  is  the  following,  but  the  enthy- 
meme in  this  case  is  in  the  minor  premise  : 

Vice  is  odious. 

Avarice  is  a  vice,  because  it  depraves. 

Therefore  avarice  is  odious. 


FOMMS  OF  SYLLOGISTIC  REASONING  201 

The  double  epicheirema  takes  the  following  form  : 

A  is  B,  because  it  is  P 
C  is  A,  because  it  is  Q 
.  • .  C  is  B. 

Or,  Man  has  a  mind,  because  he  is  rational. 

Europeans  are  men,  because  they  are  civilized. 
Therefore  Europeans  have  minds. 

The  single  epicheirema  when  resolved  or  completed  be- 
comes a  prosyllogism  and  an  episyllogism,  or  two  syllogisms. 
The  double  epicheirema  when  completed  becomes  three  syllo- 
gisms, representing  two  prosyllogisms  and  two  episyllogism s, 
one  of  the  three  being  both  a  prosyllogism  and  an  episyllo- 
gism, the  former  in  relation  to  the  following,  and  the  latter 
in  relation  to  the  preceding  syllogism.  The  student  may 
practise  their  resolution  until  familiar  with  the  processes  in- 
volved in  them.  The  resolution  of  the  double  epicheirema 
above  is  as  follows  : 

Man  has  a  mind  :  (  Whatever  is  rational  has  a  mind, 
for  he  is  rational  =  -!  Man  is  rational. 
'  Man  has  a  mind. 

Europeans  are  men  :  (  Whatever  is  civilized  is  man. 
for  they  are  civilized  =  -j  Europeans  are  civilized. 
(  Europeans  are  men. 
Therefore  Europeans  have  minds. 

The  third  syllogism  appears  in  combining  the  conclusions 
of  the  first  two  to  form  its  premises,  from  which  we  obtain 
the  conclusion  that  "  Europeans  have  minds." 

4.  Sorites. — A  sorites  is  so  called  because  the  propositions 
form  what  is  regarded  as  a  "  chain,"  or  a  continuous  series, 
of  premises  from  which  a  conclusion  is  drawn  at  the  end  of 
the  series.  It  consists  of  enthymemes  of  the  third  order,  as 
the  epicheirema  consists  of  enthymemes  of  the  first  and  sec- 
ond orders.  When  completed,  therefore,  it  forms  a  prosyllo- 
gism and  episyllogism.  To  complete  it  we  have  only  to  sup- 
ply the  intermediate  conclusions  implied  by  the  proper  prem- 


202  ELEMENTS  OF  LOGIC 

ises.     The  form  of  the  sorites  is  twofold.     It  may  be  progres- 
sive or  regressive.     The  following  are  illustrations  : 

Progressive  series. 

Bucephalus  is  a  horse. 
A  horse  is  a  quadruped, 
or,       A  quadruped  is  an  animal. 
An  animal  is  an  organism. 
.  Bucephalus  is  an  organism. 


A 

is 

B 

B 

is 

C 

C 

is 

D     ( 

D 

is 

E 

• 

•.A 

is 

E 

Regressive 

)  series. 

A 

is 

B 

C 

is 

A 

D 

is 

C 

E 

is 

D 

•.E 

is 

B 

An  animal  is  an  organism. 
A  quadruped  is  an  animal, 
or,        A  horse  is  a  quadruped. 
Bucephalus  is  a  horse. 
.  * .  Bucephalus  is  an  organism. 

The  difference  between  the  progressive  and  regressive  so- 
rites is  merely  in  the  appearance  of  the  Figure  of  the  syllo- 
gism. In  the  progressive  series  it  seems  to  be  of  the  fourth 
Figure,  and  the  propositions  are  expressed  in  that  order,  but 
the  actual  reasoning  is  performed  by  a  virtual  transmutation 
of  the  premises,  when  we  choose  the  minor  term  from  the  first 
proposition  and  the  major  term  from  the  last,  and  hence  is  of 
the  first  Figure.  In  the  regressive  series  both  the  form  of  ex- 
pression and  the  reasoning  are  in  the  first  Figure,  and  repre- 
sents the  major  premise  as  the  first  in  order. 

The  completion  of  a  sorites  may  be  illustrated  as  follows : 
The  incomplete  form  is  sometimes  called  the  occult,  and  the 
complete  the  manifest  form. 

A  is  B  Bucephalus  is  a  horse. 

B  is  C  A  horse  is  a  quadruped. 

.  • .  A  is  C  or,         .  • .  Bucephalus  is  a  quadruped. 

C  is  D  A  quadruped  is  an  animal. 

.  • .  A  is  D  .  • .  Bucephalus  is  an  animal. 

The  sorites  may  be  constructive  or  destructive.  It  is  con- 
structive when  the  premises  and  conclusion  are  affirmative, 


FORMS  OF  SYLLOGISTIC  REASONING  203 

and  hence  must  be  represented  by  the  moods  AAA,  AAI, 
and  AH,  of  the  first  Figure.  But  it  is  destructive  when  the 
conclusion  is  negative,  and  therefore  when  the  major  prem- 
ise is  negative.  The  destructive  sorites  must  then  be  rep- 
resented by  the  moods  EAE,  EAO,  and  EIO,  of  the  first 
Figure. 

There  is  a  dispute  among  logicians  as  to  whether  a  sorites 
can  exist  in  the  second  and  third  Figures.  I  agree  with  Mill 
and  Keynes  that  it  is  possible,  but  its  imjDortance  is  so  slight, 
and  its  occurrence  in  practice  so  infrequent,  that  I  do  not 
think  it  deserves  any  special  attention. 


CHAPTER  XV. 

HYPOTHETICAL    REASONING 

The  forms  of  reasoning  may  be  divided  in  the  same  manner 
as  propositions.  In  fact  they  are  determined  by  the  nature  of 
the  propositions  which  constitute  them.  We  have  already  ob- 
served that  propositions  may  be  divided  into  Categorical, 
Hypothetical,  and  Disjunctive.  Arguments  or  forms  of  rea- 
soning are  divided  according  as  the  premise  or  premises  are 
categorical,  hypothetical,  or  disjunctive.  A  categorical  syllo- 
gism is  one  in  which  all  the  propositions  are  categorical.  Of 
this  kind  we  have  already  treated,  and  it  remains  to  consider 
the  hypothetical  form. 

1.  Hypothetical  Syllogisms. — A  hypothetical  syllogism,  or 
form  of  reasoning,  is  one  in  which  one  or  more  of  the  propo- 
sitions is  hypothetical  or  conditional.  Most  frequently  it  is 
the  major  premise  alone  that  is  conditional,  while  the  minor 
premise  and  conclusion  are  categorical.  The  major  premise 
consists  of  two  propositions,  one  of  wdiich  is  dependent  upon 
the  other.  The  proposition  expressing  the  condition  is  called 
the  antecedent,  and  is  introduced  by  some  such  words  as  if, 
suppose,  alloiv,  granted  that,  'provided  that,  etc.,  all  of  which 
indicate  that  a  statement  is  made  under  conditions  restricting 
its  application.  The  projjosition  depending  upon  this  condi- 
tion is  called  the  consequent,  and  is  either  a  categorical  or  a 
disjunctive  proposition.  An  illustration  of  an  antecedent  and 
consequent  so  defined  is :  "If  the  sea  is  rough,  it  is  danger- 
ous." "If  the  sea  is  rough  "is  the  antecedent,  and  "it  is 
dangerous  "  is  the  consequent.  The  rules  regulating  the  va- 
lidity of  hypothetical  inferences  depend  upon  understanding 
the  use  of  these  terms. 

The  general  form  of  the  hypothetical  syllogism  is  that  of  an 


HYPOTHETICAL  REASONING  205 

antecedent  and  consequent,  or  a  conditional  and  a  categori- 
cal proposition  for  the  major  premise,  a  categorical  proposi- 
tion for  the  minor  premise,  and  a  categorical  proposition  for 
the  conclusion.  But  as  there  are  two  propositions  in  the 
major  premise,  the  minor  premise  may  affirm  or  deny  either 
the  antecedent  or  the  consequent.  This  gives  four  forms  to 
be  considered,  two  of  which  are  valid  and  two  of  which  are 
invalid  modes  of  reasoning.  The  two  valid  moods  are  called, 
respectively,  the  modus  ponens  and  the  modus  tollens,  or  the 
constructive  and  the  destructive  hypothetical  syllogism.  The 
modus  ponens  means  that  if  the  antecedent  be  affirmed  the 
consequent  will  follow.  The  major  premise  only  asserts  this 
relation  conditionally.  But  if  the  antecedent  be  affirmed 
categorically  in  the  minor  premise  the  conclusion  will  follow 
categorically.  The  only  effect  of  a  hypothetical  minor  premise 
is  to  make  the  conclusion  hypothetical.  The  modus  tollens 
means  that  if  the  consequent  be  denied,  the  antecedent  must 
be  denied.     The  two  forms  can  be  illustrated  as  follows : 

Modus  ponens.  Constructive  hypothetical  syllogism. 

If  A  is  B,  C  is  D  If  iron  is  impure  it  is  brittle. 

A  is  B  or,      Iron  is  impure. 

.  • .  C  is  D  .  • .  Iron  is  brittle. 

Modus  tollens.  Destructive  hypothetical  syllogism. 

If  A  is  B,  C  is  D  If  the  sun  shines  it  is  light. 

C  is  not  D  or,       It  is  not  light. 

.  * .  A  is  not  B  .  • .  The  sun  does  not  shine. 

The  rule,  therefore,  for  testing  the  validity  of  hypothetical 
reasoning  is,  that  either  the  antecedent  must  be  affirmed  or  the 
consequent  denied.  If  this  rule  be  violated  in  eitber  of  its 
conditions  a  fallacy  occurs.  This  is  illustrated  in  the  two  re- 
maining, but  invalid,  moods  of  the  hypothetical  syllogism  : 

If  A  is  B,  C  is  D  If  it  rains  it  is  cloudy. 

C  is  ~D  or,     It  is  cloudy. 

. ' .  A  is  B  .  ■ .  It  is  raining. 


206  ELEMENTS  OF  LOGIC 

This  is  a  case  of  the  fallacy  of  affirming  the  consequent.  The 
fact  that  "  A  is  B,"  or  that  "  it  is  raining,"  is  not  the  only  con- 
dition that  C  should  be  D,  or  that  it  should  be  cloudy.  Some 
other  condition  may  be  true,  so  that  if  "  A  is  B  "  follows  from 
the  affirmation  of  the  consequent,  the  other  condition  would 
follow  also.  But  there  is  nothing  to  determine  that,  and 
hence  the  known  condition  must  remain  indefinite  so  far  as 
drawing  it  from  the  truth  of  the  consequent  is  concerned. 
If  any  other  condition,  however,  may  exist  as  a  determinant 
of  the  consequent,  that  condition  may  be  the  very  opposite  of 
the  one  specified.  Thus,  C  may  be  D,  if  A  is  not  B,  or  it  may 
be  cloudy  if  it  is  not  raining,  and  in  this  case  it  will  be  ap- 
parent that  we  cannot  equally  draw  opposite  conclusions  from 
the  truth  of  the  consequent.  Hence,  we  have  no  right  to 
draw  any  inference  whatever.  A  concrete  illustration  will 
make  this  still  clearer.  "  Thus,  if  a  man's  character  be  avari- 
cious, he  will  refuse  to  give  money  for  useful  purposes  ;  but 
it  does  not  follow  that  every  person  who  refuses  to  give 
money  for  such  purposes  is  avaricious.  There  may  be  many 
proper  reasons  or  motives  leading  him  to  refuse  ;  he  may 
have  no  money,  or  he  may  consider  the  purpose  not  a  useful 
one,  or  he  may  have  more  useful  purposes  in  view."  No  in- 
ference, therefore,  can  be  drawn  from  an  affirmation  of  the 
consequent. 

A  second  fallacy  comes  from  denying  the  antecedent,  as  rep- 
resented in  the  following  form  : 

If  A  is  B,  C  is  D  If  gold  were  cheap  it  would  be  useful. 

A  is  not  B  or,  Gold  is  not  cheap. 

.  • .  C  is  not  D  .  ' .  Gold  is  not  useful. 

The  fallacy  is  due  to  the  same  causes  as  before.  The  ante- 
cedent is  not  the  only  possible  condition  of  the  consequent, 
and  hence  no  conclusion  denying  the  existence  or  truth  of  the 
consequent  can  be  drawn  from  the  denial  of  the  antecedent. 
It  is  apparent  in  the  concrete  case  that  other  qualities  besides 
cheapness  might  make  gold  useful,  and  therefore  the  absence 
of  this  quality  would  not  remove  the  usefulness  of  the  metal. 


HYPOTHETICAL  REASONING  207 

Difficulties  appear  in  some  cases  in  which  it  would  seem 
that  a  negative  conclusion  does  follow  the  denial  of  the  ante- 
cedent.    To  take  a  case  we  have  : 

If  men  are  white  they  are  Caucasians. 
Men  are  not  white. 
.  • .  They  are  not  Caucasians. 

In  supposing  whiteness  as  the  only  condition  of  being  a 
Caucasian,  as  we  generally  do,  the  absence  of  it  would  imply 
that  the  person  was  not  a  Caucasian.  And  so  perhaps  with 
the  following  instance  : 

If  fire  is  hot  it  will  burn. 
Fire  is  not  hot. 
.  • .  It  will  not  burn. 

The  usual  assumption  is  that  fire  and  the  power  to  burn  are 
the  same,  so  that  to  deny  the  heat  of  the  former  is  to  deny  the 
capacity  of  it  to  burn.  In  such  cases  a  denial  of  the  antece- 
dent seems  to  involve  an  inference  to  the  negation  of  the  con- 
sequent. 

But  we  must  not  be  deluded  by  such  instances.  They  are 
in  effect  cases  in  which  the  major  premise  is  an  exclusive  hy- 
pothetical proposition.  When  converted  it  becomes  equiva- 
leiit  to  a  modus  tollens,  in  which  the  consequent  instead  of  the 
antecedent  is  denied.  This  explains  how  the  reasoning  is 
valid.  Thus  if  whiteness  is  the  only  consideration  of  being 
Caucasian,  the  hypothetical  proposition  would  be  "  If  only 
men  are  white  they  are  Caucasian,"  which  is  equivalent  to  say- 
ing, "If  men  are  Caucasians  they  are  white."  Hence  when  we 
affirm  that  "  they  are  not  white,"  in  the  minor  premise,  we 
should  be  denying  the  consequent  and  not  the  antecedent. 
Materially,  therefore,  the  reasoning  is  correct,  and  it  would 
always  be  so  when  the  antecedent  expresses  the  only  condi- 
tion of  the  consequent.  But  formally  we  cannot  consider  it 
so,  as  there  is  nothing  in  the  form  of  the  proposition  to  indi- 
cate whether  the  antecedent  expresses  the  only  condition  or 
not.     Since  we  are  treating  only  of  formal  reasoning,  we  have 


20S  ELEMENTS  OF  LOGIC 

to  regard  all  propositions  alike  which  are  only  formal  in  their 
character.  If  the  exclusive  nature  of  the  hypothetical  propo- 
sition were  expressed,  we  might  formulate  a  rule  for  it,  but 
wThen  it  is  not  so  expressed,  we  must  consider  it  formally  as 
under  the  general  law.  Nevertheless  in  practical  reasoning 
we  should  always  be  prepared  to  distinguish  when  the  mind 
tacitly  supposes  the  material  conditions  which  might  make  the 
material  reasoning  correct  while  it  is  formally  wTrong.  In  this 
way  we  could  admit  the  truth  of  the  conclusion  and  yet  show 
that  it  has  not  been  obtained  in  accordance  with  the  formal 
law  of  hypothetical  reasoning,  or  that  we  are  liable  to  frequent 
fallacies,  if  we  allow  the  correctness  of  this  material  reasoning 
to  lead  us  into  the  indiscriminate  use  of  its  privileges  in  hy- 
pothetical syllogisms  at  large. 

There  are  three  forms  of  hypothetical  propositions  and  syllo- 
gisms which  require  notice  because  of  the  misunderstanding 
to  which  they  may  give  rise.  They  consist  of  negative  propo- 
sitions, wThile  those  we  have  illustrated  consist  of  affirmative 
propositions.  We  must  show  that  the  case  is  not  altered  by 
the  use  of  negative  propositions,  but  that  the  whole  matter 
turns  upon  the  connection  between  the  antecedent  and  the 
consequent.  There  are,  therefore,  three  more  forms  in  which 
the  major  premise  of  the  hypothetical  syllogism  may  be  ex- 
pressed.    They  are  : 

(a)  If  A  is  B,  C  is  not  D 
(6)  If  A  is  not  B,  C  is  D 
(c)  If  A  is  not  B,  C  is  not  D. 

The  peculiar  characteristic  to  be  remarked  about  these  prop- 
ositions is  their  quality  and  the  mode  of  affirming  and  denying 
the  antecedent  or  the  consequent.  In  proposition  (a)  the  an- 
tecedent will  be  treated  in  the  same  manner  as  in  previous 
instances,  but  the  consequent  will  be  affirmed  in  the  minor 
premise  by  saying  "  C  is  not  D,"  in  which  case  the  fallacy  of 
affirming  the  consequent  is  committed.  But  the  consequent 
would  be  denied  by  saying  "  C  is  D,"  and  then  we  should  be 
obliged  to  draw  the  conclusion  that  "  A  is  not  B."     In  propo- 


HYPO TH KTIC A L   REASONING  21  >9 

sition  (b)  the  consequent  being  an  affirmative  proposition 
would  be  treated  a3  before,  and  the  antecedent  would  be 
affirmed  by  making  the  minor  premise  to  be  "A  is  not  B," 
when  we  should  have  a  modus  ponens.  But  it  would  be  denied 
by  the  form  "  A  is  B,"  and  the  usual  fallacy  would  be  com- 
mitted. In  proposition  (c)  both  the  antecedent  and  the  con- 
sequent must  be  treated  as  we  have  treated  the  consequent 
in  proposition  (a)  and  the  antecedent  in  proposition  (6). 

2.  Reduction  of  Hypothetical  Syllogisms. — It  does  not  ap- 
pear from  the  manner  in  which  hypothetical  syllogisms  have 
been  discussed  that  the  process  of  reasoning  involved  can  be 
reduced  to  the  forms  of  categorical  syllogisms.  This,  however, 
is  the  fact,  and  in  order  to  understand  how  the  ordinary  laws 
of  reasoning  are  applicable  it  is  necessary  to  reduce  them  to 
the  categorical  form.  They  are  convenient,  often,  for  the  pur- 
pose of  emphasizing  the  conditional  character  of  the  major 
premise,  and  insuring  the  acceptance  of  the  conclusion  on 
those  conditions  when  the  minor  premise  is  accepted.  The 
usual  object,  however,  is  to  have  the  major  premise  accepted 
formally,  or  the  connection  between  antecedent  and  conse- 
quent, and  then  to  show  that  the  antecedent  is  true  or  the 
consequent  false,  in  order  to  obtain  a  conclusion  which  is  not 
clear  or  admitted  at  the  outset  of  the  argument. 

Nevertheless,  in  sj)ite  of  the  superior  convenience,  at  times, 
of  the  hypothetical  forms  of  reasoning,  they  can  all  be  reduced 
to  the  categorical  form.  In  all  cases  we  may  regard  the  ante- 
cedent of  the  hypothetical  major  premise  as  the  subject  of  the 
categorical  proposition,  and  the  consequent  of  the  hypothetical 
proposition  as  the  predicate  of  the  categorical.  In  some  in- 
stances this  change  is  a  very  simple  one  ;  in  others  it  can  be 
effected  only  by  a  circumlocution.  It  can  be  done  simply 
when  the  terms  of  the  antecedent  and  consequent  can  be  made 
to  form  a  phrase  representing  a  noun  and  its  modifiers. 
Thus  we  have  the  examples  : 

If  iron  is  impure  it  is  brittle.  ]     (        Impure  iron  is  brittle. 
It  is  impure.  V  -\         Iron  is  impure. 

.  •.  It  is  brittle.  ;     (  .  • .  Iron  is  brittle. 

14 


210  ELEMENTS  OF  LOGIC 

If  the  weather  is  stormy  sea  travel  will  be  dangerous. 
The  weather  is  stormy. 
.  • .  Sea  travel  will  be  dangerous. 

This  may  be  reduced  to  the  following  : 

Stormy  weather  is  a  cause  of  dangerous  sea  travel. 
The  present  weather  is  stormy. 
.  *.  The  present  weather  is  a  cause  of  dangerous  sea  travel. 

In  all  such  instances  we  practically  supply  a  new  minor  term 
in  order  to  complete  the  categorical  form,  but  it  is  only  a  par- 
ticular case  under  the  general  in  the  major  premise. 

But  all  instances  of  the  hypothetical  syllogism  are  not  so 
easily  reduced.  In  many  of  them  we  have  to  resort  to  a  cir- 
cumlocution in  such  phrases  as  "  the  case  of"  "  the  circum- 
stances that,"  etc.  Thus  in  the  hypothetical  syllogism  below 
we  must  use  this  means  of  its  conversion  : 

If  Aristotle  is  right,  slavery  is  a  proper  form  of  society. 
But  slavery  is  not  a  projDer  form  of  society. 
.'.  Aristotle  is  not  right. 

By  using  the  phrase  "the  case  of,"  this  becomes  in  the  cate- 
gorical form  : 

The  case  of  Aristotle  being  right  is  the  case  of  slavery  be- 
ing a  proper  form  of  society. 
But  slavery  is  not  a  proper  form  of  society. 
.\  Aristotle  is  not  right. 

This  is  clearly  a  syllogism  in  the  second  Figure  of  the 
mood  AEE.  It  is  evident,  therefore,  that  we  may  easily  de- 
termine the  valid  and  invalid  forms  of  hypothetical  reason- 
ing in  terms  of  the  categorical  syllogism.  We  shall  illustrate 
all  four  forms  in  order  to  make  this  clear.  First,  the  modus 
ponens : 

If  water  is  pure  it  is  good.  \     (      Pure  water  is  good. 
It  is  pure.  V   a      This  water  is  pure. 

.  •.  It  is  good.  '     '  .  \  This  water  is  good. 


HYPOTHETICAL  REASONING  211 

In  this  we  have  a  syllogism  in  Barbara,  or  AAA  of  the  first 
Figure,  and  therefore  valid. 

The  modus  tollens  will  appear  as  follows  : 

If  water  is  pure  it  is  good.  )     i     Pure  water  is  good. 
It  is  not  good.  Y  a      This  water  is  not  good. 

.\  It  is  not  pure.  )     {.:  This  water  is  not  pure. 

Here  we  have  again  a  case  of  Camestres,  or  AEE  in  the 
second  Figure,  and  valid. 

The  case  of  denying  the  antecedent  is  as  follows  : 

If  water  is  pure  it  is  good. )     (      Pure  water  is  good. 
It  is  not  pure.  >-  <      This  water  is  not  pure. 

.-.  It  is  not  good.  )     {.:  This  water  is  not  good. 

In  this  instance  we  have  a  case  of  AEE  in  the  first  Fig- 
ure, which  is  invalid.  The  fallacy  is  that  of  illicit  jwocess  of 
the  major  term.  The  major  term  is  not  distributed  in  the  ma- 
jor premise,  but  is  distributed  in  the  conclusion.  Hence, 
when  we  attempt  to  draw  a  conclusion  after  denying  the  ante- 
cedent the  fallacy  is  that  of  an  illicit  major  term. 

The  next  is  an  illustration  of  affirming  the  consequent : 

If  water  is  pure  it  is  good. )     [      Pure  water  is  good. 
It  is  good.  >-  <      This  water  is  good. 

.-.  It  is  pure.  )     ' .-.  This  water  is  pure. 

We  have  in  this  illustration  a  case  of  AAA  in  the  second 
Figure,  and  invalid  because  the  middle  term  is  undistributed. 
Hence  all  cases  of  affirming  the  consequent  are  instances  in 
which  we  commit  the  fallacy  of  illicit  process  of  the  middle 
term,  or  undistributed  middle.  The  valid  forms,  therefore, 
are  AAA  of  the  first,  and  AEE  of  the  second  Figure.  The 
invalid  forms  are  AEE  of  the  first,  and  AAA  of  the  second 
Figure. 

When  the  hypothetical  propositions  are  negative,  we  may 
either  obvert  them  into  their  corresponding  affirmatives,  or 
consider  the  invalid  forms  as  due  to  attempts  to  reason  with 
negative  premises. 


CHAPTER  XVI. 

DISJUNCTIVE   SYLLOGISMS 

A  disjunctive  syllogism  is  one  which  is  determined  by  the 
presence  of  a  disjunctive  proposition  in  one  of  its  premises, 
and  sometimes  in  the  conclusion  also.  A  disjunctive  proposi- 
tion we  have  already  learned  to  be  one  which  contains  alterna- 
tive or  mutually  exclusive  conceptions  between  which  the 
choice  of  the  mind  is  to  be  made,  and  which  are  accompanied 
by  the  disjunctives  either  and  or.  Wherever  these  terras  are 
found  in  such  propositions  they  are  meant  to  imply  that  only 
one  of  the  two  things  can  be  affirmed,  and  the  other  denied. 
Thus  when  I  say  that  "  The  weather  is  either  clear  or  cloudy," 
I  mean  that  it  cannot  be  both  at  once,  but  that  if  it  is  one  it 
cannot  be  the  other.  It  is  this  fact  which  determines  the 
right  to  draw  the  inference  in  the  disjunctive  syllogism. 

"It  is  a  disputed  question  whether  in  a  disjunctive  proposi- 
tion the  alternatives  should  be  regarded  as  in  all  cases  mu- 
tually exclusive  ;  whether,  for  example,  in  the  proposition  '  A 
is  either  B  or  C,'  it  is  necessarily  implied  that  A  cannot  be 
both  B  and  C.  There  are  really  involved  here  two  questions 
which  should  be  distinguished. 

"  (1)  In  ordinary  speech  we  do  not  intend  that  the  alterna- 
tives in  a  disjunctive  proposition  should  be  necessarily  under- 
stood as  excluding  one  another?  A  very  few  instances,  I 
think,  will  enable  us  to  decide  in  the  negative.  Take,  for  ex- 
ample, the  proposition,  '  He  has  either  used  bad  text-books,  or 
he  has  been  badly  taught : '  would  any  one  understand  this  to 
exclude  the  possibility  of  his  having  been  badly  taught  and 
having  used  bad  text-books  as  well  ?  Or,  suppose  it  laid  down 
as  a  condition  of  eligibility  for  some  appointment  that  every 
candidate  must  be  a  member  either  of  the  University  of  Ox- 


DISJUNCTIVE  SYLLOGISMS  213 

ford,  or  of  the  University  of  Cambridge,  or  of  the  University  of 
London.  Would  any  one  regard  this  as  implying  the  ineligi- 
bility of  persons  who  happened  to  be  members  of  more  than 
one  of  these  Universities  ?  Jevons  instances  the  following 
proposition,  '  A  peer  is  either  a  duke,  or  a  marquis,  or  an  earl, 
or  a  viscount,  or  a  baron.'  We  do  not  consider  this  statement 
incorrect  because  many  peers,  as  a  matter  of  fact,  possess  two 
or  more  titles. 

"  (2)  Still  this  does  not  definitely  settle  the  question.  Grant- 
ed that  in  common  speech  the  alternatives  of  a  disjunction 
may  or  may  not  be  mutually  exclusive,  it  may  neverthe- 
less be  maintained  that  this  is  only  because  common  speech  is 
elliptical,  that  in  Logic  we  should  be  more  precise,  and  that 
the  statement  '  A  is  either  B  or  C '  (where  it  may  be  both) 
should  therefore  be  written,  '  A  is  either  B  and  not  C,  or  C  and 
not  B,  or  both  B  and  C 

"  This  is  a  question  of  interpretation  or  method,  and  I  do  not 
apprehend  that  any  burning  principle  is  involved  in  the  an- 
swer that  we  may  give.  For  my  own  part  I  do  not  find  any 
sufficient  reason  for  diverging  from  the  usage  of  every-da}r  lan- 
guage. On  the  other  hand,  I  think  that  if  Logic  is  to  be  of 
practical  utility,  the  less  logical  forms  diverge  from  those  of 
ordinary  speech  the  better.  And  further,  condensed  forms 
of  expression  do  not  conduce  to  clearness,  or  even  ultimately  to 
conciseness.  For  where  our  information  is  meagre,  a  con- 
densed form  is  likely  to  express  more  than  we  intend,  and  in 
order  to  keep  within  the  mark  we  must  indicate  additional  al- 
ternatives." * 

The  purport  of  these  remarks  is  that  the  disjunction  "either 
—  or,"  is  capable  of  a  double  import.  The  first  is,  that  the 
terms  may  denote  alternatives,  either  one  of  which  may  be  suf- 
ficient to  satisfy  the  terms  of  the  proposition,  although  both 
may  exist  in  the  same  connection.  Thus  in  the  case  of  a  can- 
didate's eligibility  depending  upon  membership  in  either  the 
University  of  Oxford,  or  the  University  of  Cambridge,  etc.,  we 
mean  that  membership  in  any  one  of  them  is  sufficient,  and 
*  Keynes's  Formal  Logic,  Part  II.,  Chap.  IX.,  p.  1G7. 


214  ELEMENTS  OE  LOGIC 

that  non-membership  in  the  others  will  not  be  an  obstacle  in 
that  case.  The  second  meaning  is  that  the  two  alternatives 
shall  be  mutually  exclusive.  This  is  the  form  which  is  neces- 
sary for  correct  formal  disjunctive  reasoning,  and  as  it  fre- 
quently occurs,  we  have  to  take  it  into  account  in  a  complete 
exposition  of  the  syllogism.  The  former  case,  when  "either 
—  or"  does  not  express  mutual  exclusion  between  the  alter- 
natives, gives  rise  to  what  is  called  an  incomplete  disjunction. 
The  fallacy  incident  to  this  fact  will  be  noticed  again.  At 
present  we  have  only  to  consider  those  cases  where  "  either  — 
or  "  expresses  mutually  exclusive  alternatives.  Upon  this  as- 
sumption definite  rules  can  be  established  for  disjunctive  rea- 
soning. In  the  meantime  we  shall  use  the  terms  either  — 
or  as  the  only  accessible  symbols  for  a  formal  disjunction. 

Before  enunciating  these  laws  and  illustrating  them  we  shall 
classify  the  forms  of  the  disjunctive  syllogism.  There  are  two 
general  divisions,  the  categorical  and  the  ddemmatic,  or  the 
definite  disjunctive  syllogism  and  the  dilemma.  The  former 
consists  of  a  disjunctive  proposition  in  the  major  premise,  and 
a  categorical  in  the  minor  premise,  giving  a  categorical  conclu- 
sion. The  dilemma  consists  of  a  hypothetical  proposition  in 
the  major  premise,  and  a  disjunctive  in  the  minor  premise.  The 
subdivisions  of  these  two  general  forms  is  illustrated  in  the  fol- 
lowing: table  : 


Disjunctive  syllogisms 


Categorical    \  Modus  ponendo  tollens. 
(  Modus  tollendo  ponens. 

(  Constructive     p,     "  , " 
Dilemmatic  •<  >  ,Q.    *\    ,' 

(Destructive       (^mple.) 
v  (  Complex. 


The  first  form  of  the  categorical  disjunctive  syllogism  is 
called  the  modus  ponendo  tollens,  because  it  means  that  by 
affirming  one  of  the  alternatives  we  must  deny  the  other.  This 
is  the  meaning  of  the  Latin  phrase  denominating  it.  It  is  il- 
lustrated as  follows  : 

A  is  either  B  or  C  J         i         Oak  trees  are  either  tall  or  short. 


[    j 

A  is  not  C  )         (  .  • .  They  are  not  short. 


But  A  is  B  I       <        They  are  tall. 


DISJUNCTIVE  SYLLOGISMS  215 

The  modus  tollendo  ponens,  which  is  the  second  form,  is  so 
named  because  by  denying  one  of  the  alternatives  we  must  affirm 
the  other.     It  is  illustrated  thus  : 

A  is  either  B  or  C  )        I        The  air  is  either  cool  or  warm. 
A  is  not  B  >       S        It  is  not  cool. 

.  • .  A  is  C  )        (  .  • .  It  is  warm. 

In  these  cases  we  assume  that  the  alternatives  are  mutually 
exclusive,  and  that  the  subject  cannot  be  both  at  once,  or  that 
there  can  be  no  other  alternative.  If  this  assumption  were 
not  made  the  conclusion  would  be  invalid,  as  a  case  of  incom- 
plete disjunction.  This  is  illustrated  in  the  following  in- 
stance : 

Macaulay  either  had  great  talents  or  he  was  very  studious. 
He  had  great  talents. 
.  ■ .  He  was  not  very  studious. 

This  conclusion  does  not  necessarily  follow  because  the  alter- 
natives are  not  necessarily  exclusive  of  each  other.  A  man 
may  be  both  talented  and  studious.  Hence  when  the  disjunc- 
tion is  incomplete  in  the  major  premise,  it  gives  rise  to  a  fal- 
lacy which  is  a  petitio  principii,  and  which  will  be  explained 
again.  This  fallacy,  however,  will  not  occur,  if  we  assume  the 
disjunction  to  be  complete.  If  we  really  assume  that  Macau- 
lay  was  either  one  or  the  other  of  the  two  alternatives,  and  not 
possibly  both  of  them,  or  anything  else,  the  conclusion  is  valid. 
Very  frequently  in  such  cases  we  mean  that  the  disjunction 
shall  be  perfect,  and  hence  the  reasoning  caunot  be  criticised. 
Thus,  if  I  say  "  All  birds  are  either  white  or  black,"  and  then, 
after  affirming  that  "they  are  not  white,"  infer  that  "they  are 
black,"  I  would  be  wrong  only  because  I  was  wrong  in  the  ma- 
jor premise.  If  I  really  meant  that  these  were  the  only  two 
alternatives,  the  conclusion  would  be  true.  We  see,  therefore, 
that  there  is  no  formal  fallacy  in  disjunctive  reasoning,  but 
that  it  occurs  in  the  matter  of  the  assumption  in  the  major 
premise. 


216  ELEMENTS  OF  LOGIC 

The  laws  of  disjunctive  syllogisms  seem  to  be  quite  different 
from  the  categorical  and  the  hypothetical.  We  seem  to  infer 
a  negative  conclusion  from  affirmative  premises,  and  an  affirm- 
ative conclusion  when  one  of  the  premises  is  negative  ;  a  neg- 
ative conclusion  in  the  modus  ponendo  tollens,  and  affirmative 
in  the  modus  tollendo  ponens.  But  it  can  easily  be  shown  that 
this  is  not  exceptional.  This  can  be  done  in  two  ways.  First, 
the  major  premise,  which  is  a  disjunctive  proposition,  con- 
tains both  an  affirmative  and  a  negative  assertion,  with  the  im- 
plication that  one  is  true  and  the  other  false.  The  reasoning  is, 
therefore,  based  upon  the  law  of  contradiction  in  the  square  of 
opposition,  so  that  a  negative  conclusion  is  involved  in  the  nega- 
tion expressed,  or  implied  in  the  major  premise,  and  an  affirma- 
tive conclusion  when  the  minor  premise  is  negative.  But  the 
clearer  exposition  of  the  case  is  the  second.  As  we  have  already 
remarked  a  disjunctive  proposition  is  one  which  is  categorical 
in  its  form  and  conditional  or  hypothetical  in  its  matter.  Its 
meaning,  therefore,  must  be  determined  by  reducing  it  to  its 
equivalent,  and  we  shall  see  that  the  disjunctive  syllogism  can 
be  resolved  into  the  hypothetical,  and  this  hypothetical  into  the 
categorical,  so  that,  after  all,  the  regular  laws  of  reasoning  apply 
to  the  disjunctive  syllogism,  although  in  a  modified  and  less  ap- 
parent form. 

When  we  say  that  "  A  is  either  B  or  C,"  and  imply  that  there 
are  no  other  alternatives,  we  mean  that  if  A  is  B  it  is  not  G. 
This,  we  see,  is  a  hypothetical  proposition  with  a  negative  con- 
sequent. Or  we  may  mean  that  if  A  is  not  B  it  is  C,  in  which 
case  we  have  a  negative  antecedent.  We  have  then  only  to 
state  the  minor  premise,  as  in  the  disjunctive  syllogism,  and 
the  reasoning  becomes  hypothetical.  This  can  be  illustrated 
in  the  following  manner : 

A  is  either  BorC)     j        If  A  is  B,  it  is  not  0 
AisB  >  <        AisB 

.  • .  A  is  not  C  )     ' .  • .  A  is  not  C. 

In  the  hypothetical  form  we  have,  therefore,  a  case  of  modus 


DISJUNCTIVE  8YLL0QISM8  217 

ponens  which  is  valid.  After  the  reduction  of  the  disjunctive 
form  the  modus  fattens  appears  thus: 

A  is  either  BorC  ]     I        If  A  is  B,  it  is  not  C 
AisC  N        AisC 

.  • .  A  is  not  B  )    ' .  • .  A  is  not  B. 

But  as  the  nature  of  the  disjunctive  syllogism  is  such  that 
we  can  always  make  it  a  modus  ponens  in  the  hypothetical,  and 
as  it  is  always  formally  valid,  we  do  not  require  to  test  its 
laws  by  either  the  modus  fattens  or  the  invalid  forms  of 
hypothetical  reasoning.  It  therefore  suffices  to  convert  it 
always  into  the  one  form  for  the  purpose  of  discovering  the 
law  underlying  its  logical  process,  and  this  is  the  law  of  the 
hypothetical  syllogism,  which  we  have  already  ascertained  to 
be  the  same  as  that  of  categorical  syllogism. 

The  dilemma,  or  dilemmatic  disjunctive  syllogism,  is  subject 
to  the  laws  of  hypothetical  reasoning,  because  its  major  prem- 
ise is  hypothetical.  The  first  form  is  that  of  the  simple  con- 
structive dilemma. 

If  A  is  B,  C  is  D  ;  and  if  E  is  F,  C  is  D 
But  either  A  is  B,  or  E  is  F 
.  • .  C  is  D. 

We  observe  in  this  and  all  cases  of  the  simple  constructive 
dilemma  that  the  consequent  is  the  same  for  both  antecedents. 
This  gives  as  its  distinctive  mark  a  categorical  conclusion.  A 
concrete  illustration  is  the  following  : 

"If  a  science  furnishes  useful  facts,  it  is  worthy  of  being 
cultivated  ;  and  if  the  study  of  it  exercises  the  reasoning 
powers,  it  is  worthy  of  being  cultivated  ;  but  a  science  either 
furnishes  useful  facts,  or  its  study  exercises  the  reasoning 
powers  ;  therefore  it  is  worthy  of  being  cultivated." 

The  second  form  of  the  dilemma  is  the  complex  constructive 

dilemma. 

If  A  is  B,  C  is  D  ;  and  if  E  is  F,  G  is  H 
But  either  A  is  B,  or  E  is  F 
.  • .  Either  C  is  D,  or  G  is  H. 


218  ELEMENTS  OF  LOGIC 

This  is  different  from  the  simple  constructive  dilemma  in  that 
the  consequents  are  different,  and  this  fact  gives  as  its  dis- 
tinctive mark  a  disjunctive  conclusion.  As  an  instance  of  it 
we  have  the  following  argument : 

"  If  a  statesman  who  sees  his  former  opinions  to  he  wrong 
does  not  alter  his  course  he  is  guilty  of  deceit  ;  and  if  he  does 
alter  his  course  he  is  open  to  the  charge  of  inconsistency ; 
but  either  he  does  not  alter  his  course  or  he  does ;  therefore 
he  is  either  guilty  of  deceit  or  he  is  open  to  the  charge  of  in- 
consistency." 

The  destructive  dilemma  is  supposed  always  to  be  complex, 
because  it  can  otherwise  be  resolved  into  two  distinct  hypo- 
thetical syllogisms,  and  because  no  disjunctive  proposition 
occurs  in  it.  If  a  simple  destructive  dilemma  occurred,  it 
would  be  in  the  following  form  : 

If  A  is  B,  C  is  D  ;  and  if  E  is  F,  C  is  D 
But  C  is  not  D 
.  • .  Neither  A  is  B,  nor  E  is  F. 

This  form  of  reasoning  is  possible,  and  it  is  only  a  question  of 
definition  as  to  whether  we  shall  call  it  disjunctive  and 
dilemmatic.  But  we  should  have  to  change  the  conception  of 
"  disjunction  "  in  order  to  include  it  in  that  form.  The  com- 
plex dilemma,  therefore,  is  the  only  one  that  conrplies  with 
the  conditions.     It  is  as  follows : 

If  A  is  B,  C  is  D  ;  and  if  E  is  F,  G  is  H. 
But  either  C  is  not  D,  or  G  is  not  H. 
.  • .  Either  A  is  not  B,  or  E  is  not  F. 

A  concrete  illustration  is  found  in  the  following  argument:  "If 
this  man  were  wise,  he  would  not  speak  irreverently  of  Script- 
ure in  jest ;  and  if  he  were  good,  he  would  not  do  so  in  ear- 
nest ;  but  he  does  it  either  in  jest  or  in  earnest,  therefore  he  is 
either  not  wise  or  not  good." 

The  fallacy  incident  to  the  dilemma  is  the  same  as  in  hy- 
pothetical reasoning,  and  does  not  require  special  discussion. 


CHAPTER  XVH. 

CLASSIFICATION   OF   FALLACIES 

1st.  Definition. — The  term  "fallacy"  is  from  the  Latin 
fallo,  denoting  deception,  illusion,  error.  In  Logic  it  must  be 
distinguished  from  such  words  as  illusion.  An  illusion  is  a 
misinterpretation  of  the  data  of  sense  perception  :  a  fallacy  is 
an  error  in  reasoning.  The  term,  however,  is  often  applied 
to  those  errors  which  are  liable  to  occur  in  the  interpretation 
of  ambiguous  propositions,  made  so  by  the  displacement  of  a 
word  or  a  phrase.  But  in  the  true  logical  sense  these  errors 
are  not  fallacies.  They  may  give  rise  to  fallacies  in  reasoning 
by  rendering  the  data  uncertain  and  ambiguous,  but  they  are 
not  errors  in  reasoning  itself,  they  are  only  errors  in  inter- 
pretation. As  logicians,  however,  have  uniformly  included 
them  in  their  treatment  and  classification  of  fallacies,  we  shall 
continue  this  practice  for  the  sake  of  the  practical  convenience 
they  possess  in  ordinary  reasoning,  although  the  proper  2nace 
to  deal  with  them  is  in  Rhetoric. 

In  discussing  the  laws  of  the  syllogism  we  have  been  trying 
to  ascertain  the  rules  or  laws  which  regulate  right  reasoning. 
We  have  now  to  examine  the  illegitimate  modes  of  inference, 
or  the  mental  processes  which  result  in  fallacies,  or  erroneous 
reasoning.  We  require  some  means  of  knowing  when  the  ra- 
tional faculty  is  liable  to  go  astray,  as  well  as  when  it  has  con- 
formed to  the  true  principles  of  reasoning.  In  order  to  do 
this  we  must  classify  and  explain  the  various  forms  of  fallacy. 

2d.  Divisions. — As  already  indicated,  the  term  fallacy  is 
used  in  a  broad  sense  to  cover  both  errors  of  interpretation, 
or  of  grammatical  and  rhetorical  form,  and  errors  of  infer- 
ence, or  logical  reasoning.  This  gives  rise  to  a  twofold  di- 
vision of  fallacies,  into  Hermeneutic  and  Logical  fallacies.     I 


220  ELEMENTS   OF  LOGIC 

eruploy  the  term  "  hermeneutic "  to  denote  that  they  are 
errors  of  interpretation,  and  hence  of  the  perception  of  the 
meaning  of  a  proposition.  The  error  is  intellectual,  but  not 
ratiocinative.  This  class  of  error  or  fallacy  will  require  very 
brief  consideration.  I  recognize  but  two  forms  of  it.  First, 
the  so-called  Fallacy  of  Amphibology,  and  second,  the  Fallacy 
of  Accent. 

We  quote  the  language  of  Jevons  upon  each  of  these  forms 
of  error:  "The  Fallacy  of  Amphibology  consists  in  an  am- 
biguous grammatical  structure  of  a  sentence,  which  produce;: 
misconception.  A  celebrated  instance  occurs  in  the  prophec}r 
of  the  Spirit  in  Shakespeare's  Henry  VI.  :  '  The  Duke  yet 
lives  that  Henry  shall  depose,'  which  leaves  it  wholly  doubtful 
whether  the  Duke  shall  depose  Henry,  or  Henry  the  Duke. 
This  prophecy  is  doubtless  an  imitation  of  those  which  the 
ancient  oracle  of  Delphi  is  reported  to  have  uttered  ;  and  it 
seems  that  this  fallacy  was  a  great  resource  to  the  oracles  who 
were  not  confident  in  their  own  powers  of  foresight.  The 
Latin  language  gives  great  scope  to  misconstructions,  because 
it  does  not  require  any  fixed  order  for  the  words  of  a  sen- 
tence, and  when  there  are  two  accusative  cases  with  an  infini- 
tive verb,  it  may  be  difficult  to  tell,  except  from  the  context, 
which  comes  in  regard  to  sense  before  the  verb.  The  double 
meaning  which  may  be  given  to  '  twice  two  and  three '  arises 
from  amphibology  ;  it  may  be  7  or  10,  according  as  we  add 
the  3  after  or  before  multiplying.  In  the  careless  construc- 
tion of  sentences  it  is  often  impossible  to  tell  to  what  part  any 
adverb  or  qualifying  clause  refers.  Thus,  if  a  person  says,  '  I 
accomplished  my  business  and  returned  the  day  after,'  it  may 
be  that  the  business  was  accomplished  on  the  day  after  as 
well  as  the  return  ;  but  it  may  equally  have  been  finished  on 
the  previous  day.  Any  ambiguity  of  this  kind  may  generally 
be  avoided  by  a  simple  change  in  the  order  of  the  words  ;  as, 
for  instance,  'I  accomplished  my  business,  and  on  the  day 
after  returned.'  Amphibology  may  sometimes  arise  from 
confusing  the  subjects  and  predicates  in  a  compound  sen- 
tence, as  if  in  the  sentence,  '  Platinum  and  iron  are  very  rare 


CLASSIFICATION  OF  FALLACIES  221 

and  useful  metals,'  I  were  to  apply  the  predicate  useful  to 
platinum  and  rare  to  iron,  which  is  not  intended.  The  word 
'  respectively '  is  often  used  to  show  that  the  reader  is  not  at 
liberty  to  apply  each  predicate  to  each  subject." 

"  The  Fallacy  of  Accent  consists  in  any  ambiguity  arising 
from  a  misplaced  accent  or  emphasis  thrown  upon  some  word 
of  a  sentence.  A  ludicrous  instance  is  liable  to  occur  in  read- 
ing Chapter  XIII.  of  the  First  Book  of  Kings,  verse  27,  where 
it  is  said  of  the  prophet,  'And  he  spoke  to  his  sons,  saying, 
Saddle  me  the  ass,  and  they  saddled  him.'  The  italics  indi- 
cate that  the  word  him  was  supplied  by  the  translators  of  the 
authorized  version,  but  it  may  suggest  a  very  different  mean- 
ing. The  Commandment,  '  Thou  shalt  not  bear  false  witness 
against  thy  neighbor,'  may  be  made  by  a  slight  emphasis  of 
the  voice  on  the  last  word  to  imply  that  we  are  at  liberty  to 
bear  false  witness  against  other  persons.  Mr.  De  Morgan, 
who  remarks  this,  also  points  out  that  the  erroneous  quoting 
of  an  author,  by  unfairly  separating  a  word  from  its  context, 
or  italicising  words  which  were  not  intended  to  be  italicised, 
gives  rise  to  cases  of  this  fallacy. 

"  It  is  curious  to  observe  how  many  and  various  may  be  the 
meanings  attributable  to  the  same  sentence  according  as  em- 
phasis is  thrown  upon  one  word  or  another.  Thus  the  sen- 
tence, '  The  study  of  Logic  is  not  supposed  to  communicate  a 
knowledge  of  many  useful  facts,'  may  be  made  to  imply  that 
the  study  of  Logic  does  communicate  such  a  knowledge  al- 
though it  is  not  supposed  to  do  so  ;  or  that  it  communicates 
a  knowledge  of  a  few  useful  facts  ;  or  that  it  communicates  a 
knowledge  of  many  useless  facts.  This  ambiguity  may  be 
explained  by  considering  that  if  you  deny  a  thing  to  have  the 
group  of  qualities  A,  T3,  C,  D,  the  truth  of  your  statement  will 
be  satisfied  by  any  one  quality  being  absent,  and  an  accented 
pronunciation  will  often  be  used  to  indicate  that  which  the 
speaker  believes  to  lie  absent.  If  you  deny  that  a  particular 
fruit  is  ripe  and  sweet  and  well-flavored,  it  may  be  unripe  and 
sweet  and  well-flavored ;  or  ripe  and  sour  and  well-flavored  ; 
or  ripe  and  sweet  and  ill-flavored  ;  or  any  two  or  even  all 


222  ELEMENTS  OF  LOGIC 

three  qualities  may  be  absent.  But  if  you  deny  it  to  be  ripe 
and  sweet  and  well-flavored,  the  denial  would  be  understood 
to  refer  to  the  last  quality.  Jeremy  Bentham  was  so  much 
afraid  of  being  misled  by  this  fallacy  of  accent  that  he  em- 
ployed a  person  to  read  to  him,  as  I  have  heard,  who  had  a 
peculiarly  monotonous  manner  of  reading." 

As  already  remarked,  although  these  errors  in  the  interpre- 
tation of  propositions  are  not  strictly  fallacies,  according  to 
the  present  usual  acceptation  of  that  term,  it  may  be  well  to 
have  given  them  this  consideration,  because  they  are  often  the 
source  of  logical  fallacies  in  giving  wrong  assumptions  to  start 
from.  But  they  are  not  the  result  of  violating  any  logical  laws 
such  as  have  been  laid  down.  It  is  these  violations  with  which 
the  proper  discussion  of  fallacies  is  concerned.  Hence  we  turn 
to  the  second  class,  which  we  have  denominated  Logical  Falla- 
cies. 

Logical  Fallacies  are  errors  in  reasoning  or  inference,  and 
not  of  interpretation.  An  error  of  interpretation  is  an  error 
of  intellectual  percejjtion  ;  an  error  of  reasoning  is  an  error  of 
judgment  in  the  passage  from  one  proposition  or  conception 
to  another  assumed  to  be  contained  in  the  former.  The  data 
or  premises  may  be  correctly  interpreted  and  yet  the  infer- 
ence be  a  wrong  one.  As  an  illustration  take  the  simple  con- 
version of  propositions  in  A.  We  may  be  correct  in  our  con- 
ception of  the  proposition  "All  nations  are  aggregates  of 
men,"  but  wrong  in  the  inference  from  it,  by  simple  conver- 
sion, that  "All  aggregates  of  men  are  nations."  And  so  on 
with  all  other  forms  of  inference  where  the  conclusion  is  not 
legitimately  deduced  from  the  premises. 

Logical  fallacies  are  divided  into  formal  and  material,  ac- 
cording as  the  error  is  in  the  form  of  the  reasoning  or  in  the 
subject-matter  of  reasoning.  A  formal  fallacy  is  an  error 
which  arises  from  a  violation  of  the  formal  laws  of  inference. 
It  is  incident  to  the  mere  form  of  statement,  or,  as  it  is  often 
said,  is  a  fallacy  in  dictione  or  in  voce.  It  requires  only  a 
knowledge  of  what  the  formal  laws  of  reasoning  are  to  detect 
such  fallacies.     On  the  other  hand,  a  material  fallacy  is  one 


CLASSIFICATION  OF  FALLACIES  223 

which  is  due  to  some  peculiarity  in  the  matter  of  the  reasoning, 
and  hence  arises  independently  of  the  form  of  statement,  and 
so  is  said  to  be  extra  dictionem.  The  formal  laws  maybe  con- 
formed to,  but  owing  to  some  ambiguity  of  meaning  or  assump- 
tion of  facts  which  are  not  true  the  conclusion  may  be  materi- 
ally vitiated  in  spite  of  the  correctness  of  the  formal  reasoning. 
The  material  fallacy  can  be  detected  only  by  those  who  are 
familiar  with  the  subject-matter  of  the  discourse  or  argument. 
In  Political  Economy,  for  instance,  any  one  familiar  with  the 
laws  of  reasoning  might  be  able  to  detect  formal  errors  in 
reasoning,  but  in  order  to  discover  the  fallacies  due  to  ma- 
terial considerations,  that  is,  to  the  matter  of  the  subject,  the 
student  must  understand  Political  Economy.  It  is  the  same 
with  all  other  subjects  when  the  question  regards  material 
fallacies. 

The  further  classification  or  subdivisions  of  formal  and  ma- 
terial fallacies  must  be  considered  in  separate  paragraphs. 
We  take  up  briefly  the  formal  fallacies. 

1.  Formal  Fallacies. — These  have  been  sufficiently  defined 
as  mere  violations  of  the  principles  of  the  syllogism  which  we 
have  previously  enunciated.  They  are  determined  by  the 
number  of  terms,  the  distribution  of  terms,  and  the  nature 
of  the  premises  in  the  syllogism.  Each  species  ma}r  be  con- 
sidered briefly. 

(a)  Fallacy  of  Four  Terms,  or  Quaternio  Terminorum. — 
One  rule  of  the  syllogism  is  that  it  shall  not  contain  more 
than  three  terms  :  the  presence  of  a  fourth  term  vitiates  the 
conclusion,  because  it  prevents  that  comparison  with  a  middle 
term  which  is  necessary  to  reasoning.  A  simple  illustration 
of  the  Quaternio  Terminorum  is  the  following  : 

Men  are  mortal. 
Socrates  is  a  Greek. 
.  • .  Socrates  is  mortal. 

The  impossibility  of  drawing  an  inference  in  such  cases  is 
so  apparent,  and  the  temptation  to  do  it  is  so  unlikely  that 
errors  of  this  kind  scarcely  deserve   notice.      They  are  not 


224  ELEMENTS  OF  LOGIC 

common  enough  to  require  any  special  warning  against  them. 
It  is  only  in  the  modified  form  of  Equivocation  that  they  are 
frequent.  This  occurs  when  the  form  and  matter  of  a  term 
are  different,  that  is,  when  the  same  term  has  different  mean- 
ings. There  are,  of  course,  cases  where  the  terms  are  not 
grammatically  the  same,  but  which  are  logically  identical  in 
meaning.  These  would  only  apparently  be  cases  of  four  terms. 
The  one  circumstance  which  determines  a  case  of  four  terms 
is  a  distinction  of  material  import  that  is  not  likely  to  be  con- 
fused with  any  form  of  a  concept.  This  wiU  distinguish  such 
instances  from  Equivocation,  which  is  a  modified  form  of  Qua- 
ternio  Terminorum. 

There  is,  perhaps,  a  sense  in  which  the  fallacy  of  four  terms 
is  a  material  fallacy,  in  that  new  matter  is  introduced  into  the 
syllogism  besides  what  is  necessary  to  give  it  legitimacy.  But 
this  aspect  of  it  is  hardly  worth  serious  consideration,  al- 
though it  may  deserve  mention  for  the  purpose  of  recogniz- 
ing the  possibility. 

(6)  Illicit  Process  of  the  Middle  Term.— This  fallacy  has  al- 
ready been  explained  and  illustrated.  It  is  due  to  a  failure 
to  distribute  the  middle  term  at  least  once  in  the  premises. 
It  may  occur  in  several  ways.  One  illustration  of  it  will  suf- 
fice : 

M   =  P        Some  Pennsylvanians  are  Americans.  I  ") 

fgj  =  M       All  riiiladelphians  are  Pennsylvanians.  A  I  -pis.  I. 

.  • .  (q?\  =  P        All  Philadelphians  are  Americans.  A  j 

(c)  Illicit  Process  of  the  Major  Term. — This  is  due  to  the 
distribution  of  the  major  term  in  the  conclusion  when  it  is 
not  distributed  in  the  premises.  This  also  may  occur  in  sev- 
eral ways : 

(M)  =  P  All  men  are  mortal.  A  ] 

^   X  (M)  Some  animals  are  not  men.        O  [-  Fig.  II. 

.  •.  S  X  (p)    .*.  Some  animals  are  not  mortal.   ()  ] 

(d)  Illicit  Process  of  the  Minor  Term.— This  fallacy  is  due  to 
the  distribution  of  the  minor  term  in  the  conclusion  when  it 


CLASSIFICATION  OF  FALLACIES  225 

is  not  distributed  in  the  premises.     One  of  the  many  ways  in 
which  this  occurs  is  the  following : 

(M)  =  P  All  Germans  are  Caucasians.         A  ] 

(M)  =  S  All  Germans  are  men.  A  y  Fig.  III. 

.  • .  (S)  =  P     .  • .  All  men  are  Caucasians.  A 

(c)  Fallacy  of  Negative  Premises. — This  is  due  to  an  attempt 
to  draw  a  conclusion  when  both  premises  are  negative  and  re- 
quires no  illustration. 

(/)  Fallacy  of  Particular  Premises. — This  is  due  to  an  at- 
tempt to  reason  with  particular  premises.  This  case  when 
tested  turns  out  to  be  a  fallacy  due  to  illicit  distribution  of 
terms,  either  illicit  middle  or  illicit  major. 

We  might  also  include  in  the  formal  fallacies  breaches  of 
Rules  7  and  9  (p.  173),  or  attempts  to  draw  an  affirmative  con- 
clusion when  one  of  the  premises  is  negative,  and  to  draw  a 
universal  conclusion  when  one  of  the  premises  is  particular. 
These  errors,  however,  have  not  received  any  special  name. 

2.  Material  Fallacies. — Material  fallacies,  as  defined,  are 
due  to  something  in  the  matter  of  reasoning.  They  are,  ex- 
cepting one  instance,  the  Petitio  Principii,  cases  of  introduc- 
ing new  matter  into  the  syllogism,  while  the  form  remains  the 
same,  and  so  are  modifications  of  the  Quaternio  Terminorum. 
This  introduction  of  new  matter  may  be  either  in  the  prem- 
ises or  in  the  conclusion.  If  it  be  in  the  premises  it  must  be 
in  connection  with  the  middle  term,  which  will  give  some  form 
of  Equivocation  as  the  first  material  fallacy.  If  it  be  in  the 
conclusion,  it  must  be  in  connection  with  the  major  or  minor 
terms,  which  will  give  some  form  of  Inconsequence.  Those  of 
Equivocation  correspond  to  the  formal  fallacies  of  Undistrib- 
uted Middle  and  Quaternio  Terminorum.  Those  of  Inconse- 
quence correspond  to  the  formal  fallacies  of  Illicit  Major  and 
Illicit  Minor.  The  exceptional  case  of  Petitio  Principii,  which 
we  have  mentioned,  is  an  instance  of  assuming  matter  which  is 
not  admitted  or  not  proved.  There  seems,  therefore,  to  be 
three  forms  of  material  fallacy.  But  when  we  consider  that 
15 


226  ELEMENTS  OF  LOGIC 

fallacies  of  Inconsequence  are  assumptions  of  matter  not  in 
the  premises,  as  the  Petitio  Principii  is  an  assumption  of  un- 
proved premises,  we  may  reduce  the  last  two  to  what  may  be 
called  Fallacies  of  Presumption,  and  take  as  the  first  class 
those  of  Equivocation.  Material  fallacies  we  therefore  divide 
into  two  classes,  those  of  Equivocation  and  those  of  Pre- 
sumption. These  two  classes  require  some  further  explana- 
tion. 

The  fallacies  of  Equivocation  are  all  due  to  the  equivocal  or 
ambiguous  use  of  terms.  They  most  frequently  occur  in  con- 
nection with  the  middle  term  of  the  syllogism,  although  some 
logicians  consider  them  possible  in  connection  with  the  major 
and  minor  terms.  I  prefer  to  limit  them  to  the  middle  term 
for  the  sake  of  convenience,  although  we  might  admit  Equivo- 
cation in  the  major  and  minor  terms  as  at  the  same  time  a 
fallacy  of  Inconsequence.  The  whole  matter,  and  our  reasons 
for  the  classification,  will  not  appear  clear  until  the  material 
fallacies  have  been  explained  in  detail.  At  present  it  must 
suffice  to  obtain  their  classification. 

The  fallacy  of  Equivocation  with  logicians  generally  is  lim- 
ited to  what  is  called  Ambiguous  Middle,  and  is  not  identified 
with  the  two  fallacies  of  Accident,  and  the  two  of  Composition 
and  Division.  But  since  they  all  turn  upon  an  equivocal  use  of 
the  middle  term,  and  since,  in  my  own  experience  with  students 
of  Logic,  these  various  fallacies  are  constantly  confounded 
with  each  other,  I  am  convinced  that  they  should  be  classed 
together  under  a  common  principle.  I,  therefore,  use  the  term 
Equivocation  in  a  much  more  comprehensive  sense  than  is 
usual  with  writers  on  Logic,  and  so  to  include  the  fallacies  of 
Accident,  Simple  and  Converse,  of  Composition  and  Division, 
and  the  ordinary  case  of  Ambiguous  Middle,  which  I  shall  call 
Specific  Accident.  In  this  way  the  student  has  only  to  deter- 
mine, first,  whether  the  fallacy  turns  upon  the  use  of  an  equiv- 
ocal term,  or  iipon  the  presumption  of  matter  in  the  premises 
or  the  conclusion,  and  then  he  can  proceed  to  determine  the 
special  form  of  Equivocation.  The  peculiarities  of  these  forms 
will  be  examined  in  the  next  chapter. 


CLASSIFICATION  OF  FALLACIES 


227 


In  the  so-called  fallacies  of  Presumption,  as  already  indi- 
cated, we  may  assume  the  truth  of  the  premises  when  they 
should  be  proved,  or  we  may  assume  new  matter  in  the  con- 
clusion not  contained  in  the  premises,  which  may  be  admitted. 
This  distinction  gives  rise  to  two  general  divisions  of  the  fal- 
lacies of  Presumption,  namely,  the  Petitio  Prindpii,  or  Beg- 
ging the  Question,  and  the  Non  Sequitur,  or  False  Consequent. 
They  will  be  discussed  in  the  next  chapter.  The  following 
table  summarizes  our  classification  : 


'  Herme- 
neutic 


Si 


Logical 


Fallacy  of  Amphibology  =  Ambiguous  grammatical  structure  of  the  sen- 
tence. 
Fallacy  of  Accent  =  Ambiguity  due  to  misplaced  accent. 
'  Quaternio  Terminorum. 
Illicit  Middle. 
Illicit  Major. 
Illicit  Minor. 
Negative  Premises. 
Particular  Premises. 


Formal  - 


Equivo 
cation 


In  Quantity  {go^tion. 


( Simple  Accident. 
In  Quality  or  Accident  I  Converse  Accident. 
(Specific  Accident, 
f  Petitio     (     Petitio      fAssumptio  non  probata. 
Presump-  j  Principii  ^gnmeuti^cnl^inprobando. 

(Non  Sequitur  {gmpta^uitun. 


CHAPTER  XVHL 
MATERIAL   FALLACIES 

1st.  Fallacies  of  Equivocation. — These  have  been  de- 
fined as  caused  by  the  equivocal  use  of  terms.  I  have  divided 
them  into  two  classes,  those  of  quality,  or  Accident,  and  those 
of  quantity.  Those  of  quality  or  accident  are  so  called  because 
the  fallacy  arises  from  some  confusion  due  to  differences  of 
meaning  in  regard  to  the  attributes  denoted  by  a  term  in  a 
proposition.  Thus,  if  I  say  "  Iron  is  a  metal,"  I  affirm  "  metal " 
of  it  in  its  proper  form,  as  an  aggregate  of  certain  qualities  or 
attributes.  Now,  if  I  also  say  "  Rust  is  iron,"  I  use  the  term 
"iron "in  a  slightly  different  sense,  affirming  that  the  sub- 
stance, or  generic,  not  the  specific,  qualities  of  it  are  identical 
with  "  rust ; "  that  is  to  say  "  rust  "  is  "  iron  "  only  in  its  sub- 
stance not  in  its  form.  This  fact  prevents  me  from  drawing 
the  conclusion  that  "Rust  is  a  metal."  The  fallacies  of  quan- 
tity are  so  called  because  they  are  due  to  the  different  senses 
in  which  a  merely  numerical  aggregate  of  individuals  can  be 
taken.  Thus,  "  All  the  trees  "  may  be  taken  collectively  or  dis- 
tributively,  and  so  give  rise,  as  we  shall  see,  to  an  equivoca- 
tion. We  consider  this  form  of  fallacy  first  in  order,  and  it  is 
perhaps  the  easier  to  detect.  It  is  that  of  Composition  and 
Division. 

1.  Fallacy  of  Composition  and  Division. — Both  fallacies 
arise  from  the  confusion  of  a  collective  and  a  distributive  term, 
but  one  of  them  is  the  converse  of  the  other.  The  mode  of  de- 
termining them  can  be  expressed  in  the  following  formula  : 

~  • ,  •       (  In  the  maior  premise  the  middle  term  is  used  distributively. 

Composition  -  j      x 

|  In  the  minor  premise  the  middle  term  is  used  collectively. 

-r\-   ■  •      j  In  the  major  premise  the  middle  term  is  used  collectively. 

{  In  the  minor  premise  the  middle  term  is  used  distributively. 


MATERIAL  FALLACIES  229 

In  the  fallacy  of  Composition  it  will  thus  be  seen  that  we 
argue  from  a  distributive  to  a  collective  use  of  the  term  ; 
and,  vice  verm,  in  Division  we  argue  from  the  collective  to  the 
distributive  use  of  a  term.  Probably  a  simpler  means  of  de- 
termining the  matter  in  each  case  would  be  to  observe  wheth- 
er, as  a  whole,  the  proposition  was  used  distributively  or  col- 
lectively, and  not  to  make  the  decision  more  difficult  by  look- 
ing for  this  distinction  in  the  middle  term  of  the  premises. 

One  of  the  best  illustrations  of  a  fallacy  of  Composition  is 
the  following : 

All  the  angles  of  a  triangle  are  less  than  two  right  angles. 

A,  B,  C  are  the  angles  of  a  triangle. 

Therefore  A,  B,  C  are  less  than  two  right  angles. 

In  the  major  premise  the  proposition  is  true,  if  we  suppose 
that  the  expression  "  all  the  angles  of  a  triangle  "  is  taken  dis- 
tributively ;  that  each  angle  taken  alone  is  less  than  two  right 
angles  ;  for  taken  together  they  are  equal  to  two  right  angles. 
The  conclusion,  therefore,  cannot  be  true,  unless  A,  B,  C  are 
taken  distributively.  For  if  we  mean  in  the  major  premise 
that  each  angle  is  less  than  two  right  angles,  and  in  the  conr 
elusion  that  all  together  are  less  than  two  right  angles,  we  infer 
what  we  have  no  right  to  infer  ;  that  is,  we  argue  from  what 
is  distributively  true  of  A,  B,  C,  to  what  is  supposed  wrongly  to 
be  collectively  true  of  them.  A  similar  case  often  occurs  in 
arguments  like  the  following  : 

Thirteen  and  seventeen  are  prime  numbers. 
Thirty  is  thirteen  and  seventeen. 
Therefore  thirty  is  a  prime  number. 

In  the  major  premise  "  thirteen  and  seventeen  "  are  used  dis- 
tributively, and  in  the  minor  premise  collectively.  Thirty  not 
being  identical  with  "  thirteen  and  seventeen,"  considered  dis- 
tributively, cannot  be  identical  with  that  which  is  identical 
with  them  in  this  sense,  and  hence  the  fallacy  of  composition. 
In  the  first  illustration  the  fallacy  grows  out  of  the  ambigu- 


230  ELEMENTS  OF  LOGIC 

ous  use  of  the  word  all,  which  in  such  cases  may  have  either  a 
collective  or  a  distributive  signification.  Thus  if  we  were  to 
argue  that  because  "  All  the  peers  derive  their  titles  from  the 
crown,"  and  "  The  House  of  Parliament  consisted  of  all  the 
peers,"  therefore  "  The  House  of  Parliament  derived  its  title 
from  the  crown,"  we  should  be  committing  again  the  fallacy  of 
Composition.  "  We  must  not  argue  that  because  every  mem- 
ber of  a  jury  is  very  likely  to  judge  erroneously,  the  jury  as  a 
whole  are  very  likely  to  judge  erroneously  ;  nor  that  because 
each  of  the  witnesses  in  a  law  case  is  liable  to  give  false  or 
mistaken  evidence,  no  confidence  can  be  reposed  in  the  con- 
current testimony  of  a  number  of  witnesses."  And  we  may 
add  that  we  cannot  argue  from  the  truth  of  all  the  incidents 
in  a  story  to  the  truth  of  the  story  as  a  whole.  A  novel  may 
interweave  a  large  number  of  true  facts  and  incidents  and  yet 
not  be  true  or  historical  in  its  totality.  "  It  is  by  a  fallacy  of 
Composition  that  protective  duties  are  still  sometimes  up- 
held. Because  any  one  or  any  few  trades  which  enjoy  protec- 
tive duties  are  benefited  thereby,  it  is  supposed  that  all  trades 
at  once  might  be  benefited  similarly  ;  but  this  is  impossible, 
because  the  protection  of  one  trade  by  raising  prices  injures 
all  others." 

The  best  illustration  of  the  fallacy  of  Division  is  the  con- 
verse of  the  one  given  for  Composition.     It  is  as  follows  : 

All  the  angles  of  a  triangle  are  equal  to  two  right  angles. 

A  is  an  angle  of  a  triangle. 

Therefore  A  is  equal  to  two  right  angles. 

In  the  major  premise  the  middle  term  is  used  collectively,  as 
in  no  other  way  could  we  say  that  "all  the  angles  of  a  tri- 
angle are  equal  to  two  right  angles."  Hence  we  mean,  not 
that  each  individual  angle  is  so,  but  only  that  all  togelher  are. 
In  the  minor  premise  the  middle  term  is  distributive,  and 
hence  in  the  conclusion  we  show  that  we  have  argued  from 
wh.it  is  true  collectively,  or  of  an  aggregate,  to  what  is  true 
only  in  a  distributive  sense.     The  fallacy  is,  therefore,  one  of 


MATERIAL   FALLACIES  231 

Division.  If  I  were  to  argue  from  the  fact  that  Congress  or 
Parliament  had  voted  a  subsidy  that  Mr.  A.  or  Lord  13.  had 
voted  for  the  same,  I  should  be  committing  the  same  fallacy. 
So  also  would  be  the  argument  that  because  houses  make  a 
city  a  given  mansion  would  make  a  city  because  it  is  a  house. 
We  commit  this  fallacy  when  we  imagine  that  because  the  ag- 
gregrate  of  expense  is  large,  the  number  of  individual  items  of 
expense  will  be  large. 

2.  Fallacies  of  Accident — It  is  important  to  keep  these 
distinct  from  the  fallacies  of  Composition  and  Division.  The 
latter  have  to  do  with  numerical  or  mathematical  aggregates 
and  individuals,  the  former  with  logical  or  metaphysical  wholes 
which  represent  totals  of  attributes.  Unless  we  keep  this  in 
view  we  are  liable  to  confuse  them.  But  if  we  remember  that 
Composition  and  Division  turn  upon  the  collective  and  distrib- 
utive use  of  terms,  and  the  fallacies  of  Accident  upon  the  con- 
fusion of  essentia  and  accidentia,  or  genus  and  species  (confer- 
entia  and  differentia),  or  of  the  abstract  and  concrete,  we  shall 
have  no  difficulty  in  the  judgment  of  particular  cases.  We 
divide  the  fallacies  of  Accident  or  Quality  into  three  kinds, 
namely, 

(a)  Simjile  Accident,  or  argument  from  the  essence  or  con- 
ferentia  to  the  accident  or  differentia.  Its  dictum  in  old  Latin 
is,  a  dicto  simpliciter  ad  dictum  secundum  quid,  meaning 
"from  an  absolute  or  unconditioned  statement  to  one  which 
is  conditioned  or  accidental." 

(6)  Converse  Accident,  or  argument  from  an  accident  or  dif- 
ferentia to  the  essence  or  differentia.  Its  dictum  in  old  Latin 
is,  a  dicto  secundum  <piid  ad  dictum  simpliciter,  meaning  the 
reverse  of  that  for  Simple  Accident. 

(c)  Differential  or  Specific  Accident,  an  argument  from  ac- 
cident to  accident,  or  from  differentia  to  differentia.  Its  dic- 
tum would  be,  a  dicto  secundum  quid  ad  dictum  secundum  quid, 
meaning  "from  a  conditioned  to  a  conditioned  assertion."* 

*  This  classification  of  Ambiguous  Middle  with  the  fallacies  of  Accident 
is  entirely  new.  so  far  as  I  know,  but  I  think  the  exposition  of  it  will 
quite  justify  the  innovation. 


232  ELEMENTS  OF  LOGIC 

Jevons  defines  the  fallacy  of  Simple  Accident  to  be  an  argu- 
ment "from  a  general  rule  to  a  special  case,"  and  the  fallacy  of 
Converse  Accident,  "from  a  special  case  to  a  general  one." 
There  is  considerable  ambiguity  in  this  account  of  the  case, 
because  the  expression  "  general  rule  "  is  equivocal.  It  may 
denote  what  is  numerically  or  mathematically  "  general,"  or 
what  is  logically  "general."  In  the  former  instance  it  simply 
denotes  what  is  true  of  a  large  number  or  the  majority  of  a 
class,  but  in  the  latter  it  denotes  what  is  true  of  the  genus, 
essentia,  or  conferentia.  If  we  accept  the  former  meaning  we 
are  liable  to  confuse  the  formal  fallacy  of  an  undistributed 
middle  with  the  material  fallacy  of  accident.  Thus  if  we  were 
to  argue  that  because  "men  have  the  right  to  vote,"  and 
"  criminals  are  men  ; "  "  therefore  criminals  have  the  right  to 
vote,"  our  reasoning  would  be  perfectly  correct  as  long  as  the 
major  premise  was  regarded  as  a  universal  proposition.  But 
as  it  stands  it  is  what  is  called  a  "general "  or  indefinite  prop- 
osition, and  may  simply  denote  that  men  as  a  "general  rule" 
have  a  right  to  vote.  In  such  a  case  the  proposition  is  a  par- 
ticular one  and  the  middle  term  is  undistributed.  But  we 
should  be  arguing  from  a  "  general "  to  a  special  case,  and  yet 
the  fallacy  is  not  one  of  Accident.  We  prefer,  therefore,  to  de- 
fine the  fallacies  of  Accident  more  accurately  by  indicating 
that  the  "  general  rule  "  or  case  must  mean  the  genus,  essence, 
or  conferentia,  and  the  "  special  case  "  must  mean  the  species, 
accident,  or  differentia.  This  fallacy,  therefore,  is  an  argu- 
ment from  one  of  these  properties  or  group  of  properties  to 
the  other  ;  that  of  Simple  Accident  from  genus  to  species,  from 
conferentia  to  differentia,  from  essence  to  accident,  from  ab- 
stract to  concrete,  etc. ;  that  of  Converse  Accident  from  spe- 
cies to  genus,  etc.,  and  that  of  Differential  Accident  from  spe- 
cies to  species,  or  differentia  to  differentia,  etc.  There  can  be 
no  fallacy  in  arguing  from  genus  to  genus,  or  essence  to  es- 
sence, because  these  always  represent  the  same  or  identical 
properties. 

One  of  the  oldest  examples  of  Simple  Accident  is  the  fol- 
lowing : 


MATERIAL  FALLACIES  233 

What  you  bought  yesterday  you  eat  to-day. 
You  bought  raw  meat  yesterday. 
Therefore  you  eat  raw  meat  to-day. 

De  Morgan  humorously  remarks  of  this  ancient  illustration  : 
"This  piece  of  meat  has  remained  uncooked,  as  fresh  as  ever, 
a  prodigious  time.  It  was  raw  when  Reisch  mentioned  it  in 
the  'Margarita  Philosophica '  in  1-196  ;  and  Dr.  Whately  found 
it  in  just  the  same  state  in  1826."  It  is  not  so  accurate  an  il- 
lustration as  is  desirable  according  to  the  definition,  because 
the  subject  of  the  major  premise  is  so  indefinite,  and  is 
hardly  a  genus.  But  in  the  conclusion  the  predicate  is  as- 
serted of  the  subject,  with  the  accidental  quality  of  rawness 
added,  while  in  the  major  premise  that  predicate  is  asserted 
only  of  the  substance  or  essence  of  what  was  bought,  and 
hence  we  mistakenly  argue  from  meat  in  general,  and  without 
qualification  to  meat  in  a  particular  form.  Another  and  per- 
haps better  illustration  is  the  following  : 

Pine  wood  is  good  for  lumber. 

Matches  are  pine  wood. 

Therefore  matches  are  good  for  lumber. 

Here  the  predicate  of  the  major  premise  is  asserted  of  the 
substance  or  essence  of  "pine  wood,"  not  of  all  forms  of  it, 
while  matches  are  pine  wood  not  only  in  essence,  but  in  a  par- 
ticular form  or  accident.  We  cannot  affirm  of  this  differen- 
tial accident  what  is  true  only  of  the  essence  or  conferentia. 
So  also  we  cannot  argue  from  the  fact  that  oxygen  and  hy- 
drogen will  burn,  that  water  will  burn  because  it  is  oxygen 
and  hydrogen.  "  It  would  be  a  case  of  the  simple  fallacy  of 
Accident  to  argue  that  a  magistrate  is  justified  in  using  his 
power  to  forward  his  own  religious  views,  because  every  man 
has  a  right  to  inculcate  his  own  opinions.  Evidently  a  magis- 
trate as  a  man  has  the  rights  of  other  men,  but  in  his  capacity 
of  a  magistrate  he  is  distinguished  from  other  men,  and  he 
must  not  infer  of  his  special  powers  iD  this  respect  what  is 
true  only  of  his  rights  as  a  man."     All  fallacies  which  attempt 


284  ELEMENTS   OE   LOUK' 

the  substitution  of  a  particular  thing  for  the  generic  form  be- 
long to  this  head. 

An  illustration  of  the  fallacy  of  Converse  Accident  is  the 
following  : 

Intoxicating  liquors  act  as  a  poison. 

Wine  is  an  intoxicating  liquor. 

Therefore  wine  acts  as  a  poison. 

In  this  case  we  are  arguing  from  the  excessive  use  to  all  uses 
of  wine,  an  inference  that  is  fallacious.  The  major  premise  is 
true  only  of  a  particular  mode  of  using  liquors,  or  of  the  ex- 
cessive use  of  them,  while  the  conclusion,  unless  interpreted 
with  a  similar  qualification,  asserts  the  same  thing  of  all  forms 
of  using  them.  "  It  is  undoubtedly  true  that  to  give  to  beg- 
gars promotes  mendicancy  and  causes  evil  ;  but  if  we  inter- 
pret this  to  mean  that  assistance  is  never  to  be  given  to  those 
who  solicit  it,  we  fall  into  the  converse  fallacy  of  Accident, 
inferring  of  all  who  solicit  alms  what  is  true  only  of  those  who 
solicit  alms  as  a  profession."  Another  formulated  instance 
appears  in  the  following  illustration  : 

Loyalty  to  the  government  is  the  duty  of  all  citizens. 
Loyalty  to  Charles  I.  was  loyalty  to  the  government. 
Therefore  loyalty  to  Charles  I.  was  the  duty  of  all  citizens. 

We  may  look  at  this  instance  in  more  than  one  way.  In  the 
first  place,  the  major  premise  means  that  loyalty  is  a  duty  to 
legitimate  governments  or  to  such  as  execute  the  law,  while 
the  minor  premise  asserts  the  fact  that  loyalty  to  Charles  I. 
was  loyalty  to  the  government  whatever  its  nature  was,  and 
hence  the  conclusion  asserts  loyalty  to  Charles  I.  to  be  a  duty 
without  qualification,  and  without  distinguishing  between  him 
as  a  magistrate  and  as  a  man.  In  the  second  place,  loyalty  to 
Charles  I.  may  have  been  loyalty  to  him  as  a  private  person, 
say  by  his  servants,  while  all  citizens  could  not  be  loyal  to 
him  in  this  capacity,  and  so  it  is  an  error  to  argue  from  this 
particular  kind  of  loyalty  to  every  forni  of  it  including  civil 
allegiance. 


MATERIAL  FALLACIES  235 

When  these  fallacies  of  Accident  occur  we  may  formulate 
the  following  means  of  determining  one  form  from  the  other, 
the  simple  from  the  converse,  and  vice  versa,  The  principle 
is  the  same  as  in  the  fallacies  of  Quantity,  Composition,  and 
Division. 

I  In  the  major  premise  the  middle  term  must  be  a  genus,  es- 
sentia, or  conferentia,    the  predicate  affirmed  of  these 

~.       .      .      . ,  making  an  abstract  proposition. 

bimple  Accident <  T       .  •       .1  •,.,  .   , 

f  In  the  minor  premise  the  middle  term  must  be  a  species, 

I  accidentia,  or  differentia,  the  predicate  so  affirmed  of 

I  the  subject  making  a  concrete  proposition. 


In  the  major  premise  the  middle  term  must  be  a  S£tecies, 

accidentia,  or  differentia,   the  predicate  affirmed    of 

these  making  a  concrete  proposition. 
Converse  Accident <  r     ,-,  .     .,         .,,,    ,  ,  . 

In  the  minor  premise  the  middle  term  mnst  be   a  genus, 

essentia,  or  conferentia,  the  predicate  so  affirmed  of 

the  subject  making  an  abstract  proposition. 

These  rules  are  worded  for  the  first  Figure  of  the  syllogism, 
and  are  designed  to  indicate  a  clear  means  of  deciding  the 
case  when  it  cannot  be  done  without  such  help.  But  the 
syllogistic  form  of  inference  is  not  the  only,  and  probably  not 
the  most  frequent,  form  of  committing  this  fallacy,  although 
the  process  can  no  doubt  be  thrown  into  the  form  of  the  syl- 
logism. The  fallacy  may  often  occur  in  the  immediate  infer- 
ence by  subalternation  and  by  added  determinants  and  com- 
plex conceptions,  or  if  not  by  these,  by  a  process  which  very 
much  resembles  them.  If  we  infer  from  the  contemptible  char- 
acter of  one  "reformer"  in  a  particular  cause,  that  all  "re- 
formers "  are  bad,  we  are  committing  the  fallacy  of  converse 
Accident.  On  the  other  hand,  if  we  infer  from  the  exchangea- 
ble value  of  money,  that  the  old  Confederate  currency  has  ex- 
changeable value,  because  it  is  money,  we  commit  the  fallacy 
of  Simple  Accident. 

Without  reference  to  the  distinction  between  the  two  forms 
of  these  fallacies,  it  may  be  said  that  wherever  we  attempt  to 
make  an  interchange  of  essence  and  accidence,  or  abstract  and 
concrete,  under  the  same  term,  a  fallacy  of  Accident  is  com- 
mitted.    An  assertion  sometimes  seems  to  be  made   of  the 


236  ELEMENTS  OF  LOGIC 

whole  of  a  concrete  subject,  when  in  fact  it  is  made  only  of 
its  essential  or  of  its  accidental  forms.  Thus,  when  we  say  that 
sulphuric  acid  is  poisonous,  we  can  assert  this  predicate,  not  of 
its  essence,  but  of  a  particular  accidental  quantity  of  it,  be- 
cause it  can  be  taken  with  impunity  in  certain  forms  or 
amounts.  But  we  can  neither  argue  from  its  poisonous  char- 
acter in  large  amounts  to  its  injury  in  small  amounts,  nor  to 
its  harmfulness  in  small  quantities  from  its  dangers  in  large 
quantities.  So  of  the  assertions  that  "Governments  are  use- 
ful," "Truth  is  sublime,"  "  Charity  is  a  virtue,"  etc.  This 
will  be  the  case  in  many,  if  not  nearly  all  abstract  ideas  and 
propositions.  Indeed  we  might  say  that  we  are  extremely 
liable  to  commit  fallacies  of  Accident  in  arguing  from  the 
concrete  to  the  abstract  and  from  the  abstract  to  the  con- 
crete. We  are  certain  to  do  so  when  the  concrete  and  ab- 
stract are  viewed  logically  and  not  mathematically,  as  pre- 
viously explained.  "When  we  say  "Governments  are  useful," 
we  really  affirm  the  predicate  of  governments  in  the  abstract,  or 
perhaps  generally  of  actual  governments.  But  we  mean  usually 
to  speak  of  certain  ideal  forms  of  social  organization  and  not 
necessarily  of  any  or  all  particular  concrete  forms  of  them. 
Hence  we  speak  of  them  in  their  essence  or  conferentia,  and  so 
are  not  allowed  to  infer  by  subalternation  that  what  is  true  of 
them  in  this  sense  is  true  of  them  in  the  concrete.  On  the 
other  hand,  if  we  asserted  the  predicate  of  them  in  the  con- 
crete as  bad,  we  could  not  immediately  infer  that  the  same 
was  true  of  them  in  the  ideal  or  abstract  sense.  Although  I 
have  spoken  of  these  inferences  as  apparently  immediate,  they 
may  be  converted  into  mediate  arguments  by  supplying  a  sup- 
pressed and  perhaps  implied  premise.  Their  immediacy  ap- 
pears in  the  assumption  that  what  can  be  affirmed  of  govern- 
ment in  general  can  be  affirmed  of  governments  in  particular 
included  in  the  class. 

But  the  error  lies  precisely  in  this  assumption,  which  does 
not  allow  for  the  two  senses  in  which  the  conceptions  "  gov- 
ernment," truth,"  "  charity,"  etc.,  can  be  used.  It  is  here  that 
we  can  put  to  practical  use  the  distinction  between  logical  and 


MATERIAL  FALLACIES  237 

mathematical  generals,  or  between  the  logical  and  the  mathe- 
matical genus  ;  that  is,  between  the  genus  as  the  sum  of  the 
species,  and  the  genus  as  a  name  for  the  conferentia.  The 
latter  is  the  logical,  as  the  genus,  taken  as  the  sum  of  the  spe- 
cies, is  the  mathematical  conception  of  a  class.  If  we  say  that 
"  governments  are  useful,"  in  the  mathematical  sense,  we  mean 
all  individual  and  particular  governments  in  the  class,  and  so  no 
fallacy  will  be  committed  in  an  inference  to  the  species.  But 
if  we  say  it  in  the  logical  and  abstract  sense,  we  are  using  the 
term  in  a  sense  which  may  not  be  true  of  any  individual  case 
whatever,  and  hence  the  fallacy. 

Another  illustration  will  perhaps  make  the  matter  still 
clearer.  When  we  say  "Men  are  mortal,"  we  have  made  an  as- 
sertion which  applies  to  all  individual  men.  The  predicate  is 
asserted  mathematically  of  men.  It  is  not  asserted  of  man  in 
the  abstract,  but  of  all  men  in  the  concrete,  all  forms  and 
conditions  of  men.  Hence  when  we  form  a  minor  premise  in- 
volving any  number  or  species  of  men,  the  conclusion  follows 
necessarily  because  they  are  included  in  the  same  sense  in  the 
major  premise.  The  middle  term  has  only  a  mathematical 
signification  and  so  admits  of  no  fallacy.  But  suppose  we  af- 
firm "  Meat  is  healthy  food,"  here  is  a  statement  which  may 
be  taken  either  mathematically  to  denote  all  specific  kinds  of 
meat,  or  in  an  abstract  logical  sense  to  denote  that  the  sub- 
stance so  called  is  healthy  food,  and  so  it  would  be  spoken  of 
in  its  essence,  essential  qualities,  or  conferentia,  while  we  would 
not  intend  to  include  the  same  matter  in  its  raw  state,  its 
stale  or  decayed  condition,  or  in  unlimited  quantities.  Hence 
we  could  not  argue  from  the  universal  truth  in  the  first  case 
to  the  particular  case  in  the  second.  This  has  already  been 
illustrated  in  the  first  syllogism  representing  the  fallacy  of 
Simple  Accident.  In  fact  such  statements  are  meant  to  affirm 
the  predicate  of  certain  well-known,  perhaps  usual  and  nor- 
mal forms  of  the  subject,  and  so  exclude  the  cases  involved  in 
the  conclusion  when  the  fallacy  of  accident  is  committed. 
But  it  is  not  apparent  from  the  form  of  statement,  because 
the  mathematical  conception  of  universal  propositions  is  the 


238  ELEMENTS  <>F  LOGIC 

usual  and  the  most  natural  one.  But  it  is  just  sucli  substitu- 
tions that  the  student  must  be  on  guard  against,  as  liable,  in 
more  serious  situations  than  we  have  illustrated,  to  lead  him 
astray. 

The  technicalities  of  law  offer  a  very  rich  field  for  the  falla- 
cies of  Accident.  It  is  the  difference  between  one  case  and 
another  that  occupies  the  barrister  in  his  attempts  to  show 
that  they  are  not  included  in  the  general  rule.  In  philosophy 
the  Cartesians  committed  the  fallacy  by  denying  that  hard- 
ness, weight,  etc.,  were  essential  qualities  of  matter,  and  then 
inferring  that  a  cubic  foot  of  iron  had  no  more  matter  in  it 
than  a  cubic  foot  of  air,  because  space  or  extension  was  re- 
garded as  the  essence  of  matter.  De  Morgan  quotes  an  amus- 
ing story  from  Boccaccio  which  illustrates  the  fallacy,  but  in 
too  obtrusive  a  form  to  deceive  any  one,  and  yet  it  illustrates 
the  whole  case  : 

"A  servant  who  was  roasting  a  stork  for  his  master  was  pre- 
vailed upon  by  his  sweetheart  to  cut  off  a  leg  for  her  to  eat. 
When  the  bird  came  upon  the  table  the  master  desired  to 
know  what  had  become  of  the  other  leg.  The  man  answered 
that  storks  never  had  more  than  one  leg.  The  master,  very 
angry,  but  determined  to  strike  his  servant  dumb  before  he 
punished  him,  took  him  next  day  into  the  fields,  where  they 
saw  storks,  standing  each  on  one  leg,  as  storks  do.  The  ser- 
vant turned  triumphantly  to  his  master  ;  on  which  the  latter 
shouted,  and  the  birds  put  down  their  other  legs  and  flew 
away.  '  Ah,  sir,'  said  the  servant,  '  you  did  not  shout  to  the 
stork  at  dinner  yesterday  ;  if  you  had  done  so,  he  would  have 
shown  his  other  leg  too.'  " 

Not  all  fallacies  of  Accident  are  so  easily  detected  as  this, 
but  they  illustrate  the  same  principle  and  the  same  logical 
characteristics.  They  are  perhaps  as  frequent  as  any  other 
form  of  logical  error,  and  in  fact  the  inclination  to  make  those 
substitutions  of  two  different  things  under  the  same  name, 
and  separated  only  as  essence  and  accident,  is  so  common 
that  it  has  been  well  worth  the  pains  to  dwell  upon  the  sub- 
ject at  great  length. 


MATERIAL  FALLACIES  239 

It  remains  to  show  that  the  fallacy  of  Ambiguous  Middle, 
or  what  I  have  called  Differential  Accident,  is  rightly  included 
under  the  general  head  of  Accident.  We  have  intimated  that 
all  are  forms  of  equivocation,  that  is,  substitutions  of  one  mean- 
ing of  a  term  for  another,  and  now  we  have  to  show  that  a 
closer  relation  than  is  usually  recognized  by  logical  writers  ex- 
ists between  the  faUacies  of  Accident  and  the  ordinary  Am- 
biguous Middle.  An  illustration  will  be  tbe  best  means  of 
proving  the  case. 

The  end  of  life  is  its  perfection. 
Death  is  the  end  of  life. 
Therefore  death  is  the  perfection  of  life. 
The  ambiguity  of  the  word  "  end  "  is  perfectly  apparent,  and 
we  might  be  content  with  calling  the  fallacy  merely  one  of 
equivocation.  But  if  we  observe  closely,  although  the  word 
end  denotes  two  different  things,  there  is  a  common  idea  at 
the  basis  of  them  which  makes  the  equivocation  possible. 
This  common  characteristic  constitutes  the  generic  or  confer- 
ential  idea  of  the  word.  But  it  is  the  differential  quality 
which  in  each  case  determines  the  nature  of  the  assertion.  In 
the  major  premise  "end"  means  the  object  or  purpose  of  life 
of  which  perfection  is  asserted.  In  the  minor  premise  it 
means  the  termination  of  life,  which  is  made  identical  with 
death.  Now,  the  common  idea  or  conception  wrhich  enables 
us  to  apply  the  word  "  end  "  in  both  cases  is  the  notion  of 
limit,  or  the  point  of  interruption  in  a  line,  beyond  which  we 
need  not  go  for  a  given  purpose.  Hence  the  notion  of  object 
is  the  differentia  of  one  use,  and  termination  that  of  the  other, 
so  that  the  attempt  to  argue  from  one  to  the  other,  on  the 
ground  of  a  common  medium,  is  an  attempt  to  pass  from  one 
accident  to  another.  It  will  be  the  same  in  all  equivocal 
terms  where  the  confusion  is  not  due  to  a  mistaking  of  the  ge- 
nus or  conferentia  for  the  sjoecies  or  differentia,  and  vice  versa. 
Thus,  again  to  use  Jevons's  example, 

All  criminal  actions  ought  to  be  punished  by  law. 

Prosecutions  for  theft  are  criminal  actions. 

Therefore  prosecutions  for  theft  ought  to  be  punished  bylaw. 


240  ELEMENTS  OF  LOCK' 

Both  the  terms  "  criminal"  and  "action"  are  used  in  a  double 
sense.  In  the  major  premise  "criminal"  denot<  s  what  is  im- 
moral, and  "  action  "  a  form  of  conduct,  as  an  act  of  the  will. 
In  the  minor  premise  "criminal"  denotes  merely  pertaining  to 
a  crime,  without  implying  any  judgment  upon  its  character, 
and  "action"  denotes,  after  its  old  Latin  use,  a  suit  at  law. 
These  are  simply  differential  or  specific  meanings  of  the  term, 
which  has  no  generic  application  apart  from  such  as  arc  given, 
and  so  the  argument  is  from  one  of  these  to  the  other  through 
the  common  conception  implied  in  the  terms.  The  fallacy  is 
a  modified  form  of  Quaternio  Terminorum.  But  we  should 
call  it  the  fallacy  of  Differential  or  Specific  Accident  in  order 
to  classify  it  correctly  and  in  order  to  understand  its  charac- 
teristics. The  expression  Ambiguous  Middle  should  be  re- 
served for  a  more  comprehensive  use,  as  equivalent  to  equivo- 
cation, the  two  terms  to  be  used  interchangeably. 

2d.  Fallacies  of  Presumption. — According  to  our  pre- 
vious explanation  of  these  fallacies,  something  is  presumed  or 
assumed  which  we  have  no  right  to  take  for -ranted  in  the 
terms  of  the  syllogism.  They  are  j>resumptions  in  regard  to 
the  matter  or  contents  of  the  reasoning.  The  presumption 
may  be  regarding  the  material  truth  of  the  premises,  or  it 
may  be  regarding  the  introduction  of  new  matter  into  the 
conclusion  when  the  premises  are  admitted.  However  cor- 
rect the  formal  reasoning  may  be,  the  conclusion  may  be  viti- 
ated materially,  either  by  assuming  the  premises  when  they 
should  be  proved,  or  by  introducing  a  fourth  term  into  the 
conclusion.  We  have  then,  as  indicated  in  the  classification  of 
fallacies,  two  kinds  of  materially  false  inferences  of  Presump- 
tion, the  Petitio  Prindpii  and  the  Fallaeia  Consequentis,  or 
Non  Sequitur. 

1.  Fallacy  of  Petitio  Pklxcipii. — This  is  ordinarily  called 
Begging  the  Question,  and  means  the  assumption  of  a  fact  or 
a  premise  without  proof,  or,  as  in  the  argument  called  reason- 
ing in  a  circle,  is  an  attempt  to  prove  a  proposition  by  itself. 
This  is  a  form  of  assuming  it  when  it  shoiud  be  proved  by 
some  more  general  and  accepted  truth.     The  Petitio  Principii 


MATERIAL  FALLACIES  241 

we  divide  into  two  distinct  forms,  the  Petitio  Argumenti,  which 
is  committed  in  the  presentation  of  an  argument  or  when  at- 
tempting the  proof  of  a  jDroposition,  and  the  Ignoratio  Elenchi, 
which  is  committed  in  the  refutation  or  the  attempt  to  dis- 
prove a  proposition  ;  it  is  simply  a  little  more  complicated 
petitio  principii.  The  Petitio  Argumenti  we  again  subdivide 
into  two  forms,  the  assumptio  non  probata,  or  assumption  of 
unproved  premises,  which  ma}*  be  different  from  the  conclu- 
sion, and  the  circulus  in  probando,  or  reasoning  in  a  circle,  the 
assumption  of  premises  which  are  the  same  as  the  conclusion. 

The  assumptio  non  probata  can  be  illustrated  by  any  syllo- 
gism whatever.  Thus  if  we  were  trying  to  prove  that  "  All 
men  were  mortal,"  and  assumed  that  "  All  organic  beings  are 
mortal,"  with  the  minor  premise  that  "  All  men  are  organic 
beings,"  we  could  be  charged  with  begging  the  question  by  one 
who  did  not  admit  the  proposition  "All  organic  beings  are 
mortal."  He  might  admit  that  the  formal  reasoning  was  per- 
fectly correct,  and  that  the  conclusion  would  be  true  if  the 
premises  were  ;  but  he  woidd  insist  that  the  material  inference 
was  false  because  the  premise  was  not  admitted  or  not  proved. 
It  does  not  matter  which  premise  is  disputed,  the  effect  is  the 
same.  We  can  charge  a  petitio  principii  upon  a  man  when  we 
dispute  the  major  and  admit  the  minor  premise,  and  vice  versa, 
or  when  we  dispute  both  premises.  It  is  sufficient  to  question 
one  of  the  conditions  of  the  conclusion. 

It  is  not  merely  the  failure  to  prove  one's  premises  that  con- 
stitutes the  fallacy  of  begging  the  question.  This  failure  must 
be  one  which  occurs  when  proof  is  needed  or  demanded.  It 
is,  perhaps,  most  frequent  when  trying  to  convince  some  one 
else  of  a  given  truth,  although  it  may  occur  whenever  we 
are  trying  to  prove  to  our  own  minds  a  conclusion  without 
assuring  ourselves  sufficiently  of  the  stability  of  the  premises 
upon  which  the  conclusion  rests.  But  it  is  most  frequent  in 
arguments  with  others,  because  the  one  condition  of  proof  in 
such  cases  is  that  an  opponent  or  reader  admits  the  principles 
upon  which  the  conclusion  is  to  be  established.  "We  cannot 
prove  to  him  a  truth  with  premises  he  does  not  admit.  If  we 
16 


242  ELEMENTS  OE  LOGIO 

assume  these  without  his  acceptance,  our  reasoning  has  no 
cogency,  and  he  is  at  liberty  to  say  that  we  are  begging  the 
question,  and  this  without  disputing  either  the  formal  accu- 
racy of  our  process  or  the  truth  of  our  j>rojDOsition.  He 
merely  claims  that  the  case  is  not  proved.  A  proposition  in 
a  conclusion  may  be  true,  although  it  has  not  been  proved 
in  the  premises.  The  advantage  of  proving  it  lies  in  mak- 
ing it  a  special  case,  included  under  a  general  law  or  class, 
so  that  when  a  person  has  admitted  the  larger  he  must  per- 
force admit  the  smaller.  But  there  are  instances  in  which  we 
may  dispute  the  universality  of  a  principle  or  premises  either 
to  show  that  the  conclusion  may,  so  far  as  we  know,  be  an  ex- 
ception, or  to  assert  that  it  is  not  proved  by  such  a  case,  how- 
ever time  it  may  be  in  reality.  Suppose  we  wish  to  prove  that 
"  All  cattle  have  cloven  feet."  If,  in  order  to  do  so,  we  assert 
that  "All  ruminants  are  cloven-footed,"  and  "All  cattle  are  ru- 
minants," the  conclusion  will  follow,  provided  the  premises 
are  accepted.  But  we  can  be  charged  Avith  begging  the  ques- 
tion if  the  major  premise,  "All  ruminants  are  cloven-footed," 
is  not  true,  although  it  may  be  true  that  "  All  cattle  are  rumi- 
nants," and  also  that  they  are  all  cloven-footed.  But  the  prop- 
osition is  not  proved  except  by  the  universality  of  the  major 
premise  in  this  case.  It  is  one  thing  to  perceive  the  truth  of 
a  proposition  as  a  matter  of  fact,  and  it  is  another  to  prove  it 
by  means  of  a  higher  condition.  The  charge  of  petitio  jirinci- 
pii,  then,  must  not  be  construed  as  properly  meaning  that  the 
conclusion  is  denied,  but  only  that  it  is  not  proved.  We 
should  be  committing  a  counter-fallacy  if  we  supposed  that 
this  error  was  a  disproof  of  the  proposition  in  question. 

We  may  too  hastily  impute  the  fallacy  of  begging  the  ques- 
tion. This  is  virtually  done  when  we  demand  proof  for  a 
premise  merely  because  we  see  that  the  conclusion  must  be 
accepted  if  the  premise  is  admitted.  It  is  often  employed  in 
order  to  evade  the  issue  and  escape  conviction.  It  may  be 
permissible  sometimes  to  carry  the  demand  for  proof  back 
through  several  steps,  but  the  danger  is  that  it  will  most  fre- 
quently be  dishonestly  done,  or  be  the  mark  of  a  weak  cause. 


MATERIAL  FALLACIES  243 

De  Morgan  describes  the  case  in  the  following  language : 
"  There  is  an  opponent  fallacy  to  the  petitio  principii  which,  I 
suspect,  is  of  more  frequent  occurrence  ;  it  is  the  habit  of 
many  to  treat  an  advanced  proposition  as  a  begging  of  the 
question  the  moment  they  see  that,  if  established,  it  would 
establish  the  question.  Before  the  advancer  has  more  than 
stated  his  thesis,  and  before  he  has  had  time  to  add  that  he 
proposes  to  prove  it,  he  is  treated  as  a  sophist,  on  his  oppo- 
nent's perception  of  the  relevancy  of  his  first  step."  In  such 
emergencies  the  person  presenting  the  argument  must  ascer- 
tain whether  his  opponent  admits  in  each  case  that  the  con- 
clusion will  follow  if  the  p remises  are  true,  and  by  continuing 
this  process  he  will  either  expose  the  motive  of  his  opponent 
or  morally  weaken  his  demand  for  proof. 

This  fallacy  is  very  likely  to  occur  in  the  disjunctive  syllo- 
gism, and  especially  in  the  dilemma  ;  for  we  may  assume  the 
disjunction  to  be  complete  when  it  is  not.  There  may  be 
more  than  the  two  alternatives  usually  assumed  in  the  case. 
An  instance  of  this  occurs  in  the  sophism  which  was  used  by 
early  Greek  philosophers  to  prove  the  impossibility  of  motion. 
It  was  said  that  a  thing  must  either  move  where  it  was,  or 
where  it  was  not.  It  was  absurd  to  suppose  that  it  could 
move  where  it  was  not,  and  if  it  moved  it  could  not  be  in  the 
place  where  it  was,  and  therefore  it  was  inferred  that  its  mo- 
tion was  impossible.  But  this  conclusion,  or  the  premises 
rather,  lost  sight  of  the  third  alternative,  namely,  that  a  body 
might  move  from  the  place  where  it  was  to  a  place  where  it  was 
not,  or  had  not  been  the  moment  before.  The  omission  of 
this  alternative  in  the  premise  made  the  argument  a  petitio 
principii.  There  is  a  traditional  answer  to  this  argument 
which  we  shall  notice  under  the  Ignoratio  Elenchi. 

"Jeremy  Bentham  pointed  out  that  the  use  even  of  a  single 
name  may  imply  a  petitio  principii.  Thus  in  a  church  assem- 
bly or  synod,  where  a  discussion  is  taking  place  as  to  whether 
a  certain  doctrine  should  be  condemned,  it  would  be  a  petitio 
principii  to  argue  that  the  doctine  is  heresy,  and  therefore  it 
ought  to  be  condemned.     To  assert  that  it  is  heresy  is  to  beg 


244  ELEMENTS  OF  LOCK' 

the  question,  because  every  one  understands  by  heresy  a  doc- 
trine which  is  to  be  condemned.  Similarly,  in  Parliament,  a 
bill  is  often  opposed  on  the  ground  that  it  is  unconstitutional 
and  therefore  ought  to  be  rejected ;  but  as  no  precise  definition 
can  be  given  of  what  is  or  is  not  constitutional,  it  means  little 
more  than  that  the  measure  is  distasteful  to  the  opponent. 
Names  which  were  used  in  this  fallacious  manner  were  aptly 
called  by  Bentham  Question-begging  Epithets." 

The  cirnilus  in  pr<>b<ni<I<>  is  a  species  of  petitio  principii, 
which  consists  in  "  arguing  in  a  circle,"  or  in  assuming  as 
proof  of  a  proposition  some  assertion  which  is  identical  with 
it  in  its  import,  or  in  trying  to  prove  a  proposition  by  itself. 
Thus  to  say  that  "  Man  is  wise  because  he  is  rational,"  is  to 
argue  in  a  circle,  because  "  rational  "  is  substantially  identical 
in  meaning  with  "  wise."  So  also  would  it  be  to  argue  that 
"  The  weather  is  warm  because  it  is  summer,  and  it  is  sum- 
mer because  it  is  warm,"  and  "  Men  never  practise  excess  be- 
cause they  are  not  immoderate  in  their  habits."  Jevons's  illus- 
tration is  the  following  :  "  Consciousness  must  be  immediate 
cognition  of  an  object ;  for  I  cannot  be  said  really  to  know  a 
thing  unless  my  mind  has  been  affected  by  the  thing  itself." 
Here  "  to  know  "  and  "  immediate  cognition,"  are  identical  in 
import  and  cannot  be  used  to  prove  each  other. 

It  is  mostly  in  long  arguments  that  this  fallacy  can  be  com- 
mitted without  ready  detection.  When  we  argue  that  a  per- 
son should  submit  himself  to  the  guidance  of  his  party,  or  his 
government,  because  they  maintain  what  is  right,  and  then 
proceed  to  prove  this  by  asserting  they  are  right  because 
they  ought  to  be  submitted  to  ;  or  if  we  argued  that  lead  had 
more  matter  in  it  than  a  given  amount  of  wood,  because  it 
was  heavier,  and  that  it  was  heavier  because  it  had  more 
matter  in  it,  the  circle  would  be  so  narrow  that  it  would  be 
easy  of  detection.  But  when  the  circular  petitio  principii  oc- 
curs at  the  end  of  a  long  discourse,  as  it  often  may  do,  it  may 
be  committed  without  easy  discovery.  Only  the  closest  obser- 
vation can  secure  us  against  it.  It  is  likely  to  be  committed 
by  the  use  of  synonyms'  which  are  taken  to  express  more  than 


MATERIAL  FALLACIES  245 

the  conception  involved.  Jevons  and  Whately  have  remarked 
that  the  English  language,  being  composed  of  two  or  more 
languages,  is  liable  to  this  fallacy,  because  it  frequently  has 
several  synonymous  terms  for  the  same  conception. 

The  Ignoratio  Elenchi  is  the  second  general  class  of  fallacies 
which  we  have  included  under  the  head  of  Petitio  Principii. 
It  may  not  seem  clear  why  we  have  chosen  to  consider  it  a 
species  of  begging  the  question.  The  reason  for  so  doing  can- 
not be  fully  appreciated  until  the  fallacy  has  been  denned.  It 
has  been  called  by  Whately  and  others  the  fallacy  of  Irrele- 
vant Conclusion.  This  is  a  true  enough  description  of  it,  ex- 
cept that  the  definition  does  not  exclude  the  fattacia  consequen- 
ts or  non  sequitur.  We  prefer,  therefore,  to  define  it  as  Igno- 
rance of  the  Issue  or  Argument,  and  hence  it  consists  in  argu- 
ing to  the  wrong  point,  or  in  proving  one  thing  in  a  way  that 
seems  to  prove  something  else  ;  or  proving  something  which  is 
not  the  contradictory  of  the  thing  asserted.  It  will  be  appar- 
ent from  the  account  of  it  that  it  occurs  in  the  process  of  refu- 
tation, or  in  proving  something  which  is  supposed  to  be  the 
opposite  of  what  is  believed  or  affirmed  by  an  opponent.  We 
commit  the  fallacy  by  assuming  the  conclusion  we  reach  to 
be  in  contradiction  with  that  against  which  we  are  arguing, 
when  it  is  not  a  contradictory.  In  refutation  it  is  our  busi- 
ness to  prove  a  contradictory  of  a  given  assertion,  but  if  we 
prove  something  which  is  not  denied  by  our  opj)onent,  we  are 
evading  the  issue,  and  proving  something  that  is  irrelevant. 
Thus,  in  assuming  the  contradiction  which  is  not  a  contradic- 
tion, or  something  to  be  denied  by  our  opponent  which  is  not 
denied  by  him,  we  indirectly  beg  the  question.  This  fact  is, 
our  reason  for  classing  the  Ignoratio  Elenchi  with  the  Petitio 
Principii.  It  is  much  more  complicated  than  the  simple  case, 
but  when  we  consider  that  it  is  merely  the  counter-petition  of 
one  who  is  adducing  an  argument  in  refutation  to  that  of  the 
person  producing  proof  of  a  proposition,  we  shall  perceive 
the  right  to  regard  it  as  we  have  done.  It  is  the  assumption 
of  what  is  not  a  fact,  or  of  what  is  not  admitted  to  be  a  fact 
by  an  opponent,  and  such  an  assumption  is  of  the  nature  of 


240  ELEMENTS  OF  LOGIC 

a  petitio  jirincipii.  But  the  fact  is  concealed  arid  complicated 
by  the  circumstance  that  two  syllogisms  and  the  law  of  con- 
tradiction are  involved  in  the  explanation  of  the  case.  That 
the  issue  is  evaded  can  generally  be  determined  without  re- 
solving the  fallacy  into  a  petitio  principii.  But  it  is  this  kind 
of  fallacy  nevertheless. 

A  good  illustration  of  the  Ignoratio  Elenchi  is  the  follow- 
ing :  Suj)pose  a  man  is  accused  of  being  a  thief,  and  I  prove 
that  he  is  not  a  thief.  Now  the  proper  disproof  of  this  asser- 
tion is  to  prove  the  contradictory,  namely,  that  he  is  a  thief. 
But  if  my  opponent  instead  of  proving  this,  proves  tLat  the 
man  is  a  rogue,  he  commits  the  ignoratio  elenchi,  because  I 
have  not  denied  the  latter  proposition,  or  asserted  the  contra- 
dictory of  it.  He  virtually  begs  the  question  by  assuming 
that  to  prove  him  a  rogue  is  to  prove  him  a  thief,  and  that  he 
has  proved  the  contradictory  of  my  assertion  when  his  propo- 
sition is  not  denied.  Omitting  the  premises  which  might  be 
involved  in  establishing  either  side,  and  those  involved  in 
proving  the  proportion  assumed  to  be  disproof,  the  whole  re- 
lation may  be  represented  as  follows  : 

Proof.  Disproof.  Ignoratio  Elenchi. 


.• .  A  is  not  a  thief.         .'.  A  is  a  thief.         .*.  A  is  a  rogue. 

In  asserting  that  the  man  is  a  rogue  the  opponent  intends 
to  avail  himself  of  certain  presumptions  which  might  follow 
from  the  fact  that  the  man  was  a  rogue.  The  proof  that  A 
was  not  a  thief  might  imply,  to  untrained  minds,  that  he  was  a 
good  man,  or  the  disproof  might  be  such  as  it  was  not  easy  to 
counteract  the  effect  of.  Hence  if  the  man  can  be  proved  to 
be  a  rogue,  it  is  assumed  that  a  presumption  against  the  valid- 
ity of  the  disproof  is  established,  or  that  to  prove  him  a  rogue 
is  to  prove  him  a  thief.  The  fallacy,  nevertheless,  is  aj)parent, 
in  that  we  may  say  that  all  thieves  are  rogues,  but  not  that  all 
rogues  are  thieves.  The  fallacy,  then,  is  in  assuming  the  con- 
vertibility of  the  two  conceptions,  "  rogue  "  and  "  thief,"  and 


MATERIAL  FALLACIES  247 

that  bis  assertion  contradicts  the  proposition  involved  in  the 
original  statement. 

"  The  fallacy  is  the  great  resource  of  those  who  have  to 
support  a  weak  case.  It  is  not  unknown  in  the  legal  profes- 
sion, and  an  attorney  for  the  defendant  in  a  lawsuit  is  said  to 
have  handed  to  the  barrister  his  brief  marked,  '  No  case ; 
'abuse  the  plaintiff's  attorney.' "  In  all  the  attacks  on  a  per- 
son, or  his  character,  when  the  question  regards  a  doctrine, 
the  fallacy  is  the  same  as  in  the  case  of  the  above  attorney. 
Thus  if  I  praise  a  man's  poetry  or  his  philosophy,  it  is  no  refu- 
tation of  him  to  show  that  his  life  has  been  bad,  or  that  he 
has  lost  his  mind. 

De  Morgan  mentions  a  good  instance  :  "  If  a  man  were  to 
sue  another  for  debt,  for  goods  sold  and  delivered,  and  if  de- 
fendant were  to  reply  that  he  had  paid  for  the  goods  furnished, 
and  plaintiff  were  to  rejoin  that  he  could  find  no  record  of 
that  payment  in  his  books,  the  fallacy  would  be  probably 
committed.  The  rejoinder,  supposed  true,  shows  that  either 
defendant  has  not  paid,  or  plaintiff  keeps  negligent  accounts  ; 
and  is  a  dilemma,  one  horn  of  which  only  *  contradicts  the 
defence.  It  is  the  plaintiff's  business  to  prove  the  sale  from 
what  is  in  his  books,  not  the  absence  of  payment  from  what 
is  not ;  and  it  is  then  the  defendant's  business  to  prove  the 
payment  from  his  vouchers." 

The  observations  of  Whately  and  Mill  are  well  worth  quot- 
ing in  this  connection  at  some  length,  since  they  furnish  so 
clear  an  exposition  of  this  fallacy.  Says  the  former:  "Va- 
rious kinds  of  propositions  are,  according  to  the  occasion,  sub- 
stituted for  the  one  of  which  proof  is  required.  Sometimes 
the  particular  for  the  universal  ;  sometimes  a  proposition  with 
different  terms  ;  and  various  are  the  contrivances  employed  to 
effect  and  to  conceal  this  substitution,  and  to  make  the  conclu- 
sion which  the  sophist  has  drawn  answer  practically  the  same 
purpose  as  the  one  he  ought  to  have  established.  I  say  '  prac- 
tically the  same  purpose,'  because  it  will  often  happen  that 

*  De  Morgan  in  placing  "  only  "  in  the  position  which  it  ocenpies  in 
the  sentence  makes  his  statement  liable  to  a  fallacy  of  accent. 


24S  ELEMENTS   OF  LOGIC 

some  emotion  will  be  excited — some  sentiment  impressed  on 
the  mind — such  as  shall  bring  men  into  the  disposition  requi- 
site for  your  purpose,  though  they  may  not  have  assented  to, 
or  even  stated  distinctly  in  their  own  minds,  the  proposition 
which  it  was  your  business  to  establish.  Thus  if  a  sophist  has 
to  defend  one  who  has  been  guilty  of  some  serious  offence, 
which  he  wishes  to  extenuate,  though  he  is  unable  distinctly 
to  prove  that  it  is  not  such,  yet  if  he  can  succeed  in  making 
the  audience  laugh  ;it  some  casual  matter,  he  has  gained  prac- 
tically the  same  point. 

"So  also  if  any  one  has  pointed  out  the  extenuating  circum- 
stances in  some  particular  case  of  offence  so  as  to  show  that  it 
differs  widely  from  the  generality  of  the  same  class,  the  soph- 
ist, if  he  find  himself  unable  to  disprove  these  circumstances, 
may  do  away  with  the  force  of  them  by  simply  referring  the 
action  to  that  eery  class,  which  no  one  can  deny  that  it  belongs 
to,  and  the  very  name  of  which  will  excite  a  feeling  of  disgust 
sufficient  to  counteract  the  extenuation  ;  e.g.,  let  it  be  a  case 
of  peculation,  and  that  many  mitigating  circumstances  have 
been  brought  forward  which  cannot  be  denied  ;  the  sophisti- 
cal opponent  will  reply,  '  Well,  but  after  all,  the  man  is  a 
rogue,  and  there  is  an  end  of  it ;'  now  in  realit}'  this  was  by 
hypothesis  never  the  question  ;  and  the  mere  assertion  of 
what  was  never  denied  ought  not,  in  fairness,  to  be  regarded 
as  decisive  ;  but  practically,  the  odiousness  of  the  word,  aris- 
ing in  great  measure  from  the  association  of  those  very  circum- 
stances which  belong  to  most  of  the  class,  but  which  we  have 
supposed  to  be  absent  in  this  particular  instance,  excites  pre- 
cisely the  feeling  of  disgust  which  in  effect  destroys  the  force 
of  the  defence.  In  like  manner  we  may  refer  to  this  head  all 
cases  of  improper  appeals  to  the  passions,  and  everything  else 
which  is  mentioned  by  Aristotle  as  extraneous  to  the  matter 

in  hand  (efw  tov  7rpay/Aa.T0s). 

"  In  all  these  cases,  as  has  been  before  observed,  if  the  fal- 
lacy we  are  now  treating  of  be  employed  for  the  apparent 
establishment,  not  of  the  ultimate  conclusion,  but,  as  it  very 
commonly  happens,  of  a  premise,  then  there  will  be  a  com- 


MATERIAL  FALLACIES  249 

bination  of  this  fallacy  with  the  last  mentioned  (undue  as- 
sumption). 

"  For  instance,  instead  of  proving  that  '  this  prisoner  has  com- 
mitted an  atrocious  fraud,'  you  prove  that  '  the  fraud  he  is  ac- 
cused of  is  atrocious ; '  instead  of  proving,  as  in  the  well-known 
tale  of  Cyrus  and  the  two  coats,  that '  the  taller  boy  had  a  right 
to  force  the  other  boy  to  exchange  coats  with  him,'  you  prove 
that  '  the  exchange  would  have  been  advantageous  to  both ; ' 
instead  of  proving  that  '  a  man  has  not  a  right  to  educate  his 
children  or  dispose  of  his  property  in  the  way  he  thinks  best,' 
you  show-  that  '  the  way  in  which  he  educates  his  children  or 
disposes  of  his  property  is  not  really  the  best ;'  instead  of  prov- 
ing that '  the  poor  ought  to  be  relieved  in  this  way,'  you  prove 
that  'they  ought  to  be  relieved ;'  instead  of  j>roving  that  'an 
irrational  agent — whether  a  brute  or  a  madman — can  never  be 
deterred  from  any  act  by  the  apprehension  of  punishment,'  as, 
for  instance,  a  dog  from  sheep-biting,  by  fear  of  being  beaten, 
you  prove  that  '  the  beating  of  one  dog  does  not  operate  as  an 
example  to  other  dogs,'  etc.,  and  then  you  proceed  to  assume 
as  premises,  conclusions  different  from  what  have  really  been 
established,"  you  commit  the  fallacy  of  ignoratio  elenchi.  But 
it  is  in  a  modified  form,  because  it  appears  less  as  a  refutation 
than  as  an  attempted  confirmation  of  some  position.  They 
can,  however,  be  conceived  in  the  usual  form  by  supposing 
that  the  thesis  to  which  it  is  assumed  the  conclusions  are  op- 
posed is  suppressed.  Besides,  one  of  them,  the  instance  about 
relieving  the  poor,  might  be  considered  a  case  of  converse  acci- 
dent. But  as  something  is  proved  with  the  assumption  that 
it  is  identical  with  another  position,  while  it  is  in  reality  op- 
posed to  it,  we  have  the  ignoratio  elenchi  in  the  converse  form. 

"  A  good  instance  of  the  employment  and  exposure  of  this 
fallacy  occurs  in  Thueydides,  in  the  speeches  of  Cleon  and  Dio- 
dotus  concerning  the  Mitylenaeans  ;  the  former,  over  and  above 
his  appeal  to  the  angry  passions  of  his  audience,  urges  the  jus- 
tice of  putting  the  revolters  to  death,  which,  as  the  latter  re- 
marked, was  nothing  to  the  purpose,  since  the  Athenians  were 
not  sitting  in  judgment,  but  in  deliberation  ;  of  which  the  proper 


250  ELEMENTS  OF  I. on  If 

end  is  expediency.  And  to  prove  that  they  had  a  right  to  put 
them  to  death,  did  not  prove  this  to  he  an  advisable  step." 

Mill  ohserves  that  "  the  works  of  controversial  writers  are 
seldom  free  from  this  fallacy.  The  attempts,  for  instance,  to 
disprove  the  population  doctrines  of  Malthus  have  been  mostly 
cases  of  ignoratio  elenchi.  Malthus  has  been  supposed  to  he 
refuted  if  it  could  be  shown  that  in  some  countries  or  ages 
population  has  been  nearly  stationary  ;  as  if  he  had  asserted 
that  po2)ulation  always  increases  in  a  given  ratio,  or  had  not 
expressly  declared  that  it  increases  only  in  so  far  as  it  is  not 
restrained  by  prudence  or  kept  down  by  poverty  and  i  Ur- 
ease." 

Dr.  Johnson's  refutation  of  Berkeley's  idealism  by  kicking 
against  a  stone  is  a  similar  fallacy.  And  so  are  all  cases  of 
appeal  to  consequences  supj)osed  to  contradict  a  given  asser- 
tion before  proving  that  such  a  contradiction  exists.  In  such 
instances  the  opponent  may  accept  the  consequences,  unless 
the  contradiction  between  them  and  his  assertion  is  first 
proved. 

In  addition  to  the  general  forrn  of  the  Ignoratio  Elenchi, 
there  are  several  special  forms  which  it  is  important  in  a 
treatise  of  Logic  to  consider.  The  valid  process,  of  which  the 
ignoratio  elenchi  is  the  invalid,  is  called  the  arg amentum  ad 
rem.  The  special  invalid  forms  or  cases  of  evasion  are  the  ar- 
gumentum ad  judicium,  argumentum  adpopulum,  argument  am 
ad  hominem,  argumentum  ad  ingorantiam,  and  the  argumentum 
ad  vereeundiam. 

The  argumentum  ad  judicium  is  an  appeal  to  general  or  uni- 
versal belief,  and  so  is  based  upon  the  common  judgments  of 
mankind.  The  dictum  of  such  an  appeal  is  the  admitted  or 
assumed  truth  of  what  all  men  everywhere  believe.  The  con- 
troversialist appeals  to  this  maxim  because  he  supposes  it  is 
admitted  and  that  it  contradicts  some  conclusion  which  an  op- 
ponent is  trying  to  maintain.  Thus  if  I  deny  the  existence  of 
an  external  world,  of  spirit,  or  of  an  unseen  world,  it  would 
be  an  argumentum  ad  judicium  to  show  that  all  men  have 
evervwhere  believed  in  their  existence.     This  universal  belief 


MATERIAL  FALLACIES  251 

may  create  a  presuuij)tion  or  make  it  necessary  to  consider 
the  matter  seriously,  but  it  does  not  prove  it. 

The  argumentum  ad  populum  is  an  appeal  to  public  opinion, 
or  to  the  passions  and  prejudices  rather  than  to  the  intelli- 
gence of  people. 

The  argumentum  ad  hominem  is  an  appeal  to  the  practice, 
profession,  or  principles  of  the  person  to  whom  or  against 
whom  an  argument  is  directed.  It  is  an  effective  method  of 
silencing  an  opponent,  but  it  does  not  prove  the  case. 

The  argumentum  ad  ignorantiam  is  an  appeal  to  a  man's  ig- 
norance in  order  to  jDroduce  conviction  upon  his  inability  to 
dispute  the  case. 

The  argumentum  ad  vereeundiam  is  an  appeal  to  authority, 
or  an  accepted  body  of  doctrines. 

These  several  forms  of  argumenta  are  essentially  the  same 
in  their  principles  and  their  import.  Four  of  them  appeal  to 
certain  admitted  or  assumed  principles  which  are  supposed  to 
prove  the  case  because  they  are  assumed  to  contradict  the 
opposite,  which  is  the  position  to  be  disproved  by  the  proof  of 
its  alternative.  But  in  no  case,  unless  we  except  the  ad  homi- 
nem instance,  are  we  assured  either  that  the  dicta  upon  which 
we  depend  are  admitted  by  an  opponent,  or  that  they  are 
necessarily  contradictory  to  the  point  in  question.  They  are 
thus  evasions  of  the  issue. 

But  it  is  important  to  remark  that  they  are  not  always  ir- 
relevant or  illegitimate  merely  because  they  are  evasions. 
There  are  circumstances  in  which  it  is  perfectly  legitimate  to 
use  them  ;  only  we  must  not  suppose  that  this  legitimacy  im- 
plies that  they  are  methods  of  real  proof.  Although  they 
have  a  proper  application  they  are  not  argumenta  ad  res.  It 
is  important  to  take  this  fact  into  account  in  order  not  to 
infer,  from  their  fallacious  nature  as  arguments  to  the  point, 
that  their  illegitimacy  either  impeaches  the  proposition  in 
question  or  excludes  them  from  a  certain  relevancy  for  another 
purpose.  They  are  invalid  only  as  proofs  or  disproofs  of  a 
matter  in  discussion,  but  they  are  not  invalid  as  means  of  es- 
tablishing a  contradiction  between  two  propositions.     Com- 


252  ELEMENTS  OF  LOGIO 

mon  discourse  assumes  that  a  man  is  refuted  if  we  show 
that  he  has  contradicted  himself  and  that  he  must  accept  a 
given  conclusion  if  the  opposite  contradicts  his  profession  or 
his  practice.  But  this  is  not  the  fact.  It  does  place  him  in  a 
position  that  compels  him  to  choose  between  the  two  contra- 
dictories, but  it  does  not  decide  which  of  the  alternatives  he 
must  select.  Hence  the  charge  or  proof  of  a  contradiction  in 
a  man's  discourse  is  no  disproof  of  his  assertion,  unless  he 
still  holds  to  its  contradictory.  If  he  denies  the  contradiction 
he  may  hold  both  alternatives.  Hence  the  several  argumenta 
non  ad  res,  in  merely  proving  a  contradiction  somewhere,  are 
fallacies  of  ignoratio  elenchi,  in  the  relation  of  assuming  that 
they  prove  anything.  But  we  must  distinguish  between  this 
and  their  valid  use  for  establishing  a  contradiction.  An  illus- 
tration in  the  case  of  the  argumentum  ad  hominem  will  make 
this  position  clear.  We  quoted  the  instance  of  incomplete 
disjunction  by  the  ancient  Greek  philosophers,  who  sought  to 
prove  the  impossibility  of  motion  by  trying  to  limit  our  con- 
ception of  it  either  to  a  change  where  a  thing  is,  or  a  change 
where  it  is  not.  Tradition  has  it,  says  De  Morgan,  that  the 
originator  of  this  disjunction  called  in  a  physician  to  set  a 
dislocated  shoulder,  and  the  physician  turned  his  argument 
upon  the  philosojmer  to  prove  that  his  shoulder  was  not  hurt. 
He  argued  that  the  shoulder  must  be  put  out  of  place  either 
where  it  was,  or  where  it  was  not.  But  as  it  could  neither  be 
put  out  of  place  where  it  was,  nor  where  it  was  not,  it  could 
not  be  dislocated  at  all.  This  is  an  excellent  case  of  the  argu- 
mentum ad  hominem,  both  in  its  legitimate  and  its  illegitimate 
relation.  It  is  an  admirable  exposure  of  the  absurdity  of  the 
Greek  philosopher's  argument,  but  it  neither  disproves  the 
impossibility  of  motion  nor  proves  its  existence.  Nor  is  it  a 
refutation  of  the  assertion  which  is  imputed  by  inference  to 
the  philosopher,  namely,  that  his  shoulder  was  out  of  place. 
It  only  establishes  a  contradiction  between  his  philosophic 
doctrine  about  motion  and  his  present  belief  about  the  dislo- 
cation of  his  shoulder.  The  philosopher  would  have  only  to 
say  either  that  it  was  not  a  case  of  motion,  or  that  his  shoulder 


MATERIAL  FALLACIES  253 

was  not  displaced,  in  order  to  indicate  that  his  argument  or 
position  was  not  overthrown,  while  admitting  that  the  reason- 
ing of  the  physician  was  correct.  Nevertheless  he  would  not 
escape  the  charge  of  a  contradiction  somewhere,  and  although 
his  assertion  is  not  disproved  by  the  ad  hominem  argument, 
he  is  under  obligation  to  exjxlain  the  contradiction  or  to  give 
up  one  of  the  alternatives.  This  is  the  value  of  the  argumen- 
tum  ad  hominem,  and  of  the  other  similar  forms  of  appeal  to 
admitted  principles. 

2.  Fallacy  of  Non  Sequitub. — As  already  indicated  this  is 
generally  called  the  fallacia  consequents,  or  False  Consequent. 
It  arises  in  connection  with  the  conclusion,  and  not  in  connec- 
tion with  the  premises.  It  is,  therefore,  the  introduction  of 
new  matter  into  the  conclusion,  which  is  not  contained  in  the 
premises.  There  is  no  special  necessity  for  subdividing  it 
into  distinct  forms,  except  that  one  class  has  received  a  separ- 
ate name  for  the  sake  of  particular  convenience,  and  perhaps 
because  of  its  peculiar  frequency.  If  we  must  distinguish 
them  at  all,  it  must  be  into  the  common  non  sequitur,  and  the 
non  causa  pro  causa,  or  false  cause,  often  called  the  post  hoc, 
ergo  propter  hoc,  fallacy.  The  form  in  which  the  fallacy  usu- 
ally occurs  can  be  represented  in  the  following  manner : 

All  men  are  rational. 
Socrates  is  a  man. 
Therefore  Socrates  is  noble. 

It  is  evident  that  this  conclusion  cannot  follow  from  the  prem- 
ises unless  we  regard  "noble"  as  identical  with  "rational," 
which  it  is  not  intended  to  be.  The  fourth  term  is  here  in  the 
conclusion.  De  Morgan's  illustration  of  the  fallacy  is  less 
simple.     It  is  : 

Episcopacy  is  of  Scripture  origin. 

The  Church  of  England  is  the  only  Episcopal  church  in 

England. 
Therefore  the  church  established  is  the  church  that  should 

be  supported. 


254  ELEMENTS  OF  LOGIC 

It  is  evident  that  nothing  has  been  said  about  supporting 
the  church  in  the  premises,  and  hence  it  does  not  follow  from 
them.  The  fallacy  is  determined  wholly  by  the  presence  of 
new  matter  in  the  conclusion.  It  closely  resembles  the  formal 
fallacies  of  illicit  major  and  minor.  The  difference  is  that  in 
the  latter  the  addition  is  quantitative,  while  in  the  non  sequitur 
it  is  qualitative. 

The  literal  meaning  of  non  sequitur  would  apply  to  any  fal- 
lacy whatever,  because  the  fallacy  means  that  the  conclusion 
does  not  follow  from  the  premises.  But  technically  logicians 
meau  or  should  mean  by  this  particular  term  that  the  conclu- 
sion does  not  follow  from  the  premises,  although  they  are 
true.  The  petitio  principii  vitiates  the  conclusion  because  of 
false  premises  ;  in  the  non  sequitur  the  premises  are  not  dis- 
puted, but  are  admitted,  at  least  for  the  sake  of  the  argument. 
It  is  important  to  observe  in  this  connection  that  both  falla- 
cies are  possible  at  the  same  time  and  in  the  same  syllogism. 
We  may  question  the  premises  and  so  charge  a  petitio  principii 
upon  the  conclusion  ;  or  we  may  say  that  even  if  the  premises 
are  true  the  conclusion  does  not  follow,  in  which  case  we  im- 
pute a  non  sequitur  to  the  reasoning.  Therefore  whenever  we 
can  make  the  error  turn  upon  false  assumptions  in  the  prem- 
ises, we  charge  the  former  fallacy  against  the  reasoning,  and 
whenever  it  turns  upon  false  assumption  in  the  conclusion,  in- 
dependently of  the  premises,  we  charge  the  latter  fallacy. 

Very  frequently  the  fallacy  of  non  sequitur  is  due  to  appar- 
ent cases  of  immediate  reasoning,  which  are  in  reality  enthy- 
memes.  Thus  if  we  were  to  say  "  History  is  authentic  because 
mankind  has  accepted  its  statements,"  or  "  Philosophy  is  use- 
less because  it  bakes  no  bread,"  we  might  be  charged  with  a 
non  sequitur  on  the  ground  that  the  conclusion  was  not  in- 
cluded in  the  premise.  But  since  the  argument  is  an  enthy- 
meme  we  can  complete  it  in  the  usual  way,  so  that  the  conclu- 
sion after  all  might  be  included  in  the  terms  of  the  suppressed 
premise.  Thus  the  major  premise  of  the  first  enthymeme  is 
"  Whatever  mankind  has  accepted  is  authentic,"  and  of  the 
second,  "  Whatever  bakes  no  bread  is  useless."     When  these 


MATERIAL   FALLACIES  255 

are  supplied  we  find  that  the  conclusion  is  valid  unless  we 
can  impeach  the  premises,  but  to  question  thein  turns  the  fal- 
lacy into  a  petitio  principii.  We  thus  discover  that  what  may 
be  regarded  as  a  non  sequitur  in  one  relation  may  be  a.  petitio 
principii  in  another  ;  what  is  not  involved  in  one  premise  may 
be  begged  in  the  other. 

We  have  therefore  to  be  careful  in  deciding  when  a  fallacy 
is  a  non  sequitur  alone.  The  pure  and  simple  form  of  it  oc- 
curs when  both  premises  are  admitted,  either  in  reality  or  for 
the  sake  of  argument.  In  such  cases  as  we  have  just  indi- 
cated it  coincides  with  the  petitio  principii,  and  may  be  re- 
duced to  it.  But  when  it  occurs  in  its  pure  form  this  cannot 
be  done. 

The  fallacy  of  False  Cause,  or  non  causa  pro  causa,  is  the  mis- 
take of  imagining  a  necessary  connection  where  there  is  none, 
or  of  confusing  a  causal  connection  with  a  mere  coexistence  or 
sequence.  It  occurs  when  we  argue  that  a  certain  thiug  is 
the  cause  of  another  when  we  find  them  occurring  together. 
Thus  if  we  were  to  argue  that  a  change  of  the  weather  was 
due  to  the  occurrence  of  a  new  or  full  moon,  because  they 
coincided,  or  because  the  former  immediately  followed  the 
latter  ;  or  if  we  attributed  a  pestilence  to  the  occurrence  of 
a  comet ;  or  a  death  in  the  family  to  an  eclipse  of  the  sun,  we 
should  be  committing  this  fallacy.  The  Latin  phrase,  post  hoc, 
ergo  propter  hoc,  indicates  the  manner  in  which  the  conclu- 
sion is  drawn,  and  upon  what  it  depends.  "  AVhen  things 
are  seen  together,"  says  De  Morgan,  "  there  is  frequently  an 
assumption  of  necessary  connection.  There  is,  of  course,  a 
presumption  of  connection  :  if  A  and  B  have  never  been  seen 
apart,  there  is  probability  (the  amount  of  which  depends  upon 
the  number  of  instances  observed)  that  the  removal  of  one 
would  be  the  removal  of  the  other.  It  is  when  there  is  only 
one  instance  to  proceed  upon  that  the  assumption  falls  under 
this  fallacy  ;  were  there  but  two,  inductive  probability  might 
be  said  to  begin.  The  fallacy  could  then  consist  only  in  es- 
timating the  probability  too  high."  But  a  probability  is  no 
proof.     The    inference  may  be  a  deductive    fallacy,  however 


256  ELEMENTS  OF  LOGIC 

great  the  probability,  and  in  spite  of  the  inference  being  in- 
ductively legitimate.  No  number  of  mere  coexistences  or  se- 
quences contains  the  statement  of  the  cause  of  phenomena, 
and  we  are  not  entitled  to  infer  it  from  them.  Necessary  con- 
nection is  not  involved  in  the  mere  fact  of  connection.  If  it 
were,  I  might  argue  that  night  was  the  cause  of  day,  or  ri<-r 
versa,  because  we  find  that  one  invariably  precedes  the  other  ; 
or  I  might  argue  that  the  flight  of  birds  was  the  cause  of 
springtime,  because  it  accompanies  the  latter. 

If  we  analyze  the  cases  of  non  causa  pro  causa,  however,  we 
shall  find  that  they  too  may  coincide  with  a  petitio  principii, 
and  perhaps  they  should  be  classified  with  that  form  of  fallacy. 
Thus  we  might  say  that  the  inference  that  night  was  the  cause 
of  day  was  a  non  sequitur  when  drawn  from  their  invariable 
connection,  but  when  we  complete  the  syllogism  by  supplying 
the  suppressed  premise  the  major  premise  would  be,  "  All  that 
precedes  day  is  the  cause  of  it."  The  minor  premise  would  be, 
"  Night  precedes  day,"  and  the  conclusion  would  follow  as  in- 
volved in  the  major,  although  not  in  the  minor,  premise.  But 
we  may  charge  the  major  premise  with  begging  the  question, 
and  hence,  as  before,  this  case,  which  appears  a  non  sequitur  in 
relation  to  the  minor  premise,  is  a  petitio  principii  in  relation 
to  the  major  premise.  All  post  hoc,  ergo  propter  hoc,  fallacies 
can  be  reduced  in  this  way,  and  hence  it  might  seem  best  to 
include  them  as  a  species  of  begging  the  question.  But  as  they 
usually  occur  with  an  enthymeme  where  the  conclusion  in 
such  cases  is  not  included  in  the  premise,  the  conveniences  of 
controversy  make  it  best  to  regard  the  fallacy  as  a  non  sequitur, 
although  it  is  one  which  coincides  with  a  petitio  principii,  or 
may  so  coincide  with  it.  'But  the  most  perfect  form  of  non 
sequitur  will  occur  when  both  premises  are  unquestionable. 

3d.  General  Observations. — The  first  observation  to  be 
made  regarding  the  fallacies  which  we  have  just  considered  is 
that  they  are  not  always  distinct  from  each  other.  This  is 
apparent  in  the  fact  that  the  last  two  often  coincide,  and  that 
the  non  causa  pro  causa  may  be  resolved  into  a  petitio  principii 
when  the  suppressed  premise  is  supplied.     A  similar  reduction 


MATERIAL   FALLACIES  257 

might  be  possible  with  some  of  the  others.  For  example,  take 
the  fallacies  of  Accident,  and  in  particular  the  illustration  of 
the  use  of  pine  wood  : 

Pine  wood  is  good  for  lumber. 

Matches  are  pine  wood. 

Therefore  matches  are  good  for  lumber. 

We  gave  this  as  a  fallacy  of  Accident.  But  in  fact  it  may  be 
resolved  in  two  other  ways  at  the  same  time,  which  may  show 
why  the  fallacy  of  Accident  occurs.  In  the  first  place,  the 
major  premise  is  an  indefinite  or  general  j>roposition,  and  we 
have  already  learned  that  such  propositions  are  very  frequently 
particular  in  their  real  import.  This  is,  in  fact,  the  real  mean- 
ing of  the  statement.  It  is  not  true,  nor  would  it  be  intended 
to  assert,  that  all  pine  wood,  that  is,  all  forms  of  it,  are  good 
for  lumber,  but  only  that  some  pine  wood  is  good  for  lumber. 
But  thus  to  convert  the  real  import  of  the  major  premise  into 
a  particular  proposition,  making  the  syllogism  IAA  of  the 
first  Figure,  prevents  the  distribution  of  the  middle  term,  so 
that  the  fallacy  woidd  virtually  be  a  formal  one.  Many  of  the 
fallacies  of  Accident  can  be  so  reduced.  But  it  is  only  because 
we  are  viewing  the  premises  in  their  quantitative  signification 
instead  of  their  qualitative.  For  it  is  true,  qualitatively,  that 
"All  pine  wood  is  good  for  lumber,"  that  is,  in  substance, 
but  not  in  every  form,  and  hence  the  case  of  Accident  can  be 
brought  against  this  conception  of  it.  But  interpreting  the 
case  mathematically,  what  would  be  regarded  logically  and 
qualitatively  a  fallacy  of  Accident  becomes  formally  and  quan- 
titatively an  illicit  middle. 

In  the  second  place,  since  we  suppose  the  material  meaning 
of  the  major  premise  to  be  that  "  Some  pine  wood  is  good  for 
lumber,"  we  impeach  the  truth  of  the  proposition  universally, 
and  it  is  upon  its  universal  truth  that  the  conclusion  depends. 
Hence,  in  considering  the  premise  or  premises  doubtful  we 
can  regard  the  fallacy  as  a  pelitio  prindpii.  There  are  thus 
two  fallacies,  one  formal  and  the  other  material,  which  can  be 
17 


258  ELEMENTS  OF  LOGIC 

imputed  to  this  syllogism,  besides  that  of  Accident.  In  fact 
we  can  make  it  a  fallacy  of  Accident  only  upon  the  supposition 
that  the  major  premise  is  universally  true  of  the  essentia  of  pine 
wood,  but  not  of  its  accidentia,  while  the  predicate  of  the  con- 
clusion would  connect  with  one  of  its  accidents  what  had  been 
connected  in  the  major  premise  only  with  the  essence.  But 
aside  from  this  interpretation  either  an  illicit  middle  or  a 
petitio  principii  can  be  imputed  to  the  syllogism. 

Perhaps  a  similar  resolution  of  Conrposition  and  Division 
could  be  made,  because  the  premises  in  syllogisms  committing 
those  fallacies  are  capable  of  a  double  interpretation.  It  is 
sufficient  to  suggest  the  possibility,  and  the  actual  achieve- 
ment of  it  can  be  left  to  the  student.  And  it  will  not  be  nec- 
essary to  say  more  on  the  close  relation  between  the  non  sequi- 
tur  and  the  petitio  principii  in  many  cases,  after  having  shown 
that  the  two  may  be  applied  to  the  same  conclusion,  but  in 
different  relations,  one  of  them  indicating  assent  to  a  premise, 
but  not  to  the  inference,  and  the  other  indicating  that  one  of 
the  premises  vitiates  it.  The  two  will  not  coincide  when  they 
are  imputable,  one  of  them  only  to  false  premises  and  the 
other  only  to  a  false  inference. 

One  more  remark,  which  has  been  alluded  to,  it  is  impor- 
tant to  make.  The  imputation  of  a  fallacy  in  the  reasoning 
does  not  necessarily  imply  that  the  proposition  in  the  conclu- 
sion is  a  false  one.  In  many  cases  the  falsehood  of  the  propo- 
sition and  the  existence  of  the  fallacy  go  together  ;  but  it  is 
not  always  the  fact,  and  we  must  learn  to  recognize  this  fact 
because  although  a  proposition  may  be  true,  it  may  lead  to 
error  to  have  it  connected  falsely  with  another  proposition  as 
proof  when  that  proposition  may  not  be  true,  or  when  the 
conclusion  is  not  an  inference  from  it.  We  commit  a  fallacy 
when  we  suppose  that  an  error  in  reasoning  is  a  sufficient  dis- 
proof of  a  proposition.  The  fallacy  thus  committed  is  an  igno- 
ratio  elenchi.  All  that  the  existence  of  a  fallacy  can  establish 
is  a  mistake  in  the  mode  of  proving  a  proposition,  unless  it 
serve  as  the  means  of  discovering  the  actual  error  in  our 
propositions.     We  usually  discover  the  error,  in  fact,  before 


MATERIAL  FALLACIES  259 

we  find  why  or  how  it  has  been  committed.  But  the  fallacy 
in  reasoning  is  an  error  growing  mainly  out  of  an  attempt  to 
deduce  one  connection  of  terms  from  another,  and  so  will  not 
always  be  an  index  of  material  errors  of  fact.  They  are,  of 
course,  accompanied  by  error  somewhere  of  a  material  kind 
frequently  enough,  but  not  necessarily  implying  it  where  the 
ordinary  mind  assumes  it ;  we  require  to  be  on  our  guard 
against  committing  a  fallacy  when  imputing  one  to  others. 
We  must  always  distinguish  between  the  error  in  fact  which 
we  may  first  perceive,  and  the  error  in  reasoning  to  which 
such  a  discovery  may  have  led  us. 

Another  important  remark  is  that  it  is  not  necessary  to  put 
the  argument  into  the  form  of  a  syllogism  in  order  to  discover 
what  the  fallacy  is.  We  have  only  to  observe  the  manner  of 
substituting  one  term  for  another.  Most  frequently  in  actual 
discourse  arguments  are  either  stated  in  the  form  of  enthy- 
niemes,  or  the  premises  are  so  expressed  as  to  effectually  con- 
ceal the  Mood  and  Figure  of  the  syllogism,  and  we  are  left  en- 
tirely to  depend  upon  the  manner  in  which  we  use  certain 
terms.  Then,  since  in  enthymemes  we  can  construct  them 
into  syllogisms,  at  least,  of  the  first  or  of  the  second  Figure, 
as  we  please,  in  one  of  which  the  same  matter  may  be  valid 
which  is  invalid  in  the  other,  and  since  the  three  Figures  can 
be  reduced  to  the  first  at  pleasure,  it  will  not  be  necessary  to 
consider  the  form  in  detail,  but  only  how  we  substitute  one 
term  for  another.  If  the  fallacy  be  a  formal  one,  it  will  be 
most  easily  detected  in  some  cases  by  observing  the  form  of 
the  argument,  but  in  some  cases  this  is  not  necessary.  Be- 
sides formal  fallacies  are  not  so  often  committed  as  the  mate- 
rial. When  any  doubt,  however,  exists  about  the  nature  of 
an  illegitimate  inference,  it  is  best  to  throw  the  argument  into 
the  form  of  a  syllogism,  and  then  ascertain  its  relation  to  the 
general  rules. 

But  in  many,  if  not  in  most  instances  of  material  fallacy,  we 
can  determine  the  error  by  observing  the  two  or  more  senses 
in  which  a  term  is  used  without  stopping  to  consider  whether 
the  form  of  the  syllogism  is  expressed  correctly  or  not,  be- 


260  ELEMENTS  OF  LOGIC 

cause  it  may  either  be  thought  in  a  different  manner  from  the 
expressed  relation,  or  it  may  be  materially  what  it  is  not  for- 
mally.    Thus  in  the  following  syllogism  : 

White  men  are  Caucasians. 
The  Germans  are  Caucasians. 
Therefore  the  Germans  are  white. 

we  should  be  guilty  of  a  formal  fallacy  of  illicit  middle.  But 
since  we  may  have  stated  the  major  premise  in  a  form  in  which 
it  was  not  thought,  namely,  in  the  inverse  form,  the  proposi- 
tion being  a  definition,  the  reasoning  may  be  in  the  form  of 
the  first  instead  of  the  second  Figure,  as  it  is  stated,  and 
hence  perfectly  valid.  We  cannot,  of  course,  always  rely 
upon  this  method  of  dealing  with  an  argument,  but  in  cases 
of  material  reasoning  we  either  use  the  first  Figure  most  com- 
monly, unless  we  are  proving  a  negative,  or  our  data  can  be 
so  easily  reduced  to  it,  that  we  can  generally  depend  upon  the 
mere  form  of  substitution  of  one  term  for  another  in  order  to 
determine  the  nature  of  the  fallacy.  Thus  if  we  try  to  prove 
that  a  man  should  give  alms  to  a  particular  person  on  the 
ground  of  his  duty  to  be  charitable,  we  commit  a  fallacy  of  Ac- 
cident, because  we  argue  from  the  genus  "  charity  "  to  a  par- 
ticular case  of  it  where  an  accidental  or  differential  circum- 
stance may  modify  the  obligation.  Similarly  with  the  fallacies 
of  Quantity,  if  I  argue  from  the  effect  of  a  forest  in  producing 
a  thick  shade  to  a  similar  effect  from  a  single  tree  I  commit  the 
fallacy  of  Composition.  We  have  seen  how  the  non  sequitur 
may  be  imputed  without  considering,  at  least  in  some  cases, 
more  than  the  statement  which  is  assigned  as  its  ground,  al- 
though a  completion  of  the  argument  by  suj>plying  a  sup- 
dressed  premise  may  convert  it  into  a  petitio  principii.  This 
is  only  an  illustration  of  observing  whether  the  conclusion  is 
deduced  from  the  given  data  or  not.  In  the  ignoratio  elenchi  we 
never  require  to  construct  the  syllogism,  but  only  to  observe 
whether  the  conclusion  evades  the  question  or  not,  or  whether 
the  assumed  contradiction  is  a  true  one  or  not.     Since  all 


MATERIAL  FALLACIES  2G1 

material  fallacies,  with  the  possible  exception  of  the  pelitio 
principii  in  some  cases,  are  a  modified  form  of  Quaternio 
TerminoiTim,  we  have  only  to  see  whether  terms  are  used 
throughout  an  argument  in  an  identical  sense  or  not,  in 
order  to  determine  the  nature  of  the  fallacy  committed  in  any 
particular  case.* 

*  On  Fallacies  consult  De  "Morgan  :  Formal  Logic,  Chapter  XIII.  ;  Mill : 
Logic,  Book  V.,  especially  Chapters  V.,  VI..  and  VII.  ;  Whately:  Ele- 
ments of  Logic,  Book  III. ;  Hamilton :  Lectures  on  Logic,  Lecture  XXIII. 


CHAPTER  XIX. 

QUANTIFICATION   OF   THE   PREDICATE 

A  treatise  on  Logic  is  hardly  complete  that  omits  an  ac- 
count of  recent  doctrines  regarding  what  is  called  the  "  quan- 
tification of  the  predicate."  The  usual  expositions  of  the 
subject  are  confined  to  the  forms  left  by  Aristotle,  and  which, 
sufficing  for  practical  purposes,  are  best  adapted  to  the  actual 
usages  of  language.  But  language  does  not  always  express 
explicitly  what  thought  involves  implicitly,  and  hence  many 
logicians  have  felt  it  necessary  to  correct  this  defect  by  an 
ideal  scheme  of  logical  doctrine  which  might  enable  us  better 
to  understand  logical  processes  in  their  pure  forms,  and  then 
to  modify  this  scheme  to  suit  the  exigencies  of  defective  usage. 
Sir  William  Hamilton,  Professor  De  Morgan,  and  George 
Bentham,  all  about  the  same  time  conceived  the  propriety,  or 
at  least  the  possibility,  of  modifying  logical  doctrine  by  the 
"  quantification  of  the  predicate."  This  view  we  shall  pro- 
ceed to  explain,  with  its  importance  in  practical  reasoning. 

We  have  already  shown  what  the  quantification  of  the  subject 
is,  although  it  has  not  been  stated  under  that  name.  But  to 
quantify  it  is  only  to  say  whether  the  whole  or  the  part  of  it 
is  taken  into  account.  Its  quantification  refers  to  its  distri- 
bution or  non-distribution.  Hence  to  quantify  the  predicate 
is  to  state  whether  the  whole  or  only  the  part  of  it  agrees 
with  the  subject,  or  differs  from  it.  We  have  seen  that  the 
proposition  "  Men  are  wise,"  is  an  indefinite  one  so  far  as  its 
explicit  statement  is  concerned,  and  that  we  cannot  deal  with 
it  logically  or  with  any  degree  of  assurance  as  to  the  results 
unless  we  first  know  whether  it  means  "  all  men  "  or  "  some 
men  are  wise."     This  is  quantifying  the  subject  in  order  to 


QUANTIFICATION  OF   THE  PREDICATE  263 

bring  the  proposition  into  a  definite  form  for  logical  use. 
Thus  far  the  ordinary  Logic  proceeds,  but  no  further.  But 
why  not  also  quantify  the  predicate  in  a  similar  manner,  in 
order  to  evade  the  equivocations  incident  to  its  indefinite 
forms?  In  the  above  proposition  it  is  said  to  be  undistributed, 
because  nothing  is  stated  or  implied  to  indicate  whether  the 
whole  or  only  a  part  of  its  extension  is  taken  into  account. 
It  is,  therefore,  perfectly  indefinite.  But  in  some  cases,  in 
spite  of  this  mode  of  statement,  we  happen  to  know  that  the 
meaning  of  the  predicate  is  definite ;  that  it  is  identical  with 
the  subject  in  its  quantity.  As  already  explained  this  is  the 
case  with  all  definitions  in  which  subject  and  predicate  are  con- 
vertible terms.  Supposing  that  we  define  man  as  a  rational 
animal,  we  can  say  with  equal  truth  "  All  men  are  rational 
animals,"  and  "  All  rational  animals  are  men,"  and  so  with  any 
other  definition.  The  same  happens  to  be  true  of  the  prop- 
osition "  White  men  are  Caucasians."  We  can  convert  it  sim- 
ply into  "All  Caucasians  are  white,"  but  only  because  we 
happen  to  know  that  whiteness  arid  Caucasian  are  convertible 
terms.  According  to  the  formal  laws  enunciated  regarding 
the  form  of  such  propositions,  it  would  have  to  be  converted 
per  accidens  :  so  also  even  with  definitions.  But  as  the  form 
of  a  definition  and  that  of  an  ordinary  assertion  cannot  be 
distinguished  in  general  usage,  why  would  it  not  be  an  im- 
provement in  the  science  of  Logic  to  treat  the  predicate  as  we 
have  treated  the  subject,  and  to  state  explicitly  what  is  implic- 
itly involved  in  it?  The  answer  to  this  question  will  appear 
in  the  sequel,  after  we  have  shown  how  the  quantification  can 
be  effected  and  what  are  its  consequences  to  the  ordinary 
forms  of  reasoning.  Jevons's  exposition  suffices  for  the  pur- 
pose. 

"  In  the  proposition  '  All  metals  are  elements,'  the  subject 
is  quantified,  but  the  predicate  is  uot ;  we  know  that  all  metals 
are  elements,  but  the  proposition  does  not  distinctly  assert 
whether  metals  make  the  whole  of  the  elements  or  not.  In 
the  quantified  proposition  'All  metals  are  some  elements,'  the 
little  word  some  expresses  clearly  that  in  reality  the  metals 


2C4  ELEMENTS  OF  LOGIC 

form  only  a  part  of  the  elements.*  Aristotle  avoided  the  use 
of  any  mark  of  quantity  by  assuming,  as  we  have  seen,  that  all 
affirmative  propositions  have  a  particular  predicate,  like  the 
example  just  given ;  and  that  only  negative  propositions  have  a 
distributed  or  universal  predicate.  The  fact  is,  however,  that 
he  was  entirely  in  error,  and  thus  excluded  from  his  system 
an  infinite  number  of  affirmative  propositions  which  are  uni- 
versal in  both  terms.  It  is  true  that  '  All  equilateral  triangles 
are  all  equiangular  triangles,'  but  this  proposition  could  not 
have  appeared  in  his  system  excej)t  in  the  mutilated  form,  'All 
equilateral  triangles  are  equiangular.'  Such  a  proposition  as 
'  London  is  the  capital  of  England,'  or  '  Iron  is  the  cheapest 
metal,'  had  no  proper  place  whatever  in  his  syllogism,  since 
both  terms  are  singular  and  identical  with  each  other,  and 
both  are  accordingly  universal. 

"As  soon  as  we  allow  the  quantity  of  the  predicate  to  be 
stated  the  forms  of  reasoning  become  much  simplified.  We 
may  first  consider  the  process  of  conversion.  In  our  dis- 
cussion of  the  subject  it  was  necessary  to  distinguish  between 
simple  conversion  and  conversion  by  limitation.  But  now  one 
simple  process  of  simple  conversion  is  sufficient  for  all  kinds 
of  propositions.  Thus  the  quantified  proposition  of  the  form 
A,  '  All  metals  are  some  elements,'  would  be  simply  converted 
into  '  some  elements  are  all  metals.'  "  The  quantified  form  of 
A,  "  All  metals  are  all  elements,"  would  be  simply  converted 
into  "  All  elements  are  all  metals  ;"  and  so  on  with  all  propo- 
sitions. We  could  simply  proceed  upon  the  rule  that  what- 
ever we  do  with  one  term  we  could  do  with  the  other.  Their 
meaning  is  made  definite  by  their  explicit  quantification. 

"  The  doubly  universal  proposition  is  of  most  frequent  oc- 

*  If  the  ordinary  canon  about  the  signification  of  some,  as  previously 
defined  (p.  116),  is  to  be  enforced  here,  Jevons  is  wrong  in  saying  that  the 
word  denotes  only  a  part.  It  merely  asserts  distinctly  a  part,  and  does 
not  assert  or  even  imply  anything  about  the  whole.  But  it  does  indicate 
that  we  are  not  to  assume  anything  about  the  whole  of  the  predicate.  If 
we  adopt  its  use  with  the  implication  that  it  denotes  only  a  part,  we 
should  have  to  define  it  so. 


QUANTIFICATION  OF  THE  PREDICATE  265 

currence,  as  in  the  case  of  all  definitions  and  singular  prop- 
ositions. I  may  give  as  instances,  '  Honesty  is  the  best 
policy,'  '  The  greatest  truths  are  the  simplest  truths,'  '  Vir- 
tue alone  is  happiness  below,'  '  Self-exaltation  is  the  fool's 
paradise.' 

"When  affirmative  propositions  are  expressed  in  the  quanti- 
fied form  all  immediate  inferences  can  be  readily  drawn  from 
them  by  this  one  rule,  that  whatever  ive  do  with  one  term  toe 
should  do  with  the  other  term.  Thus,  from  the  doubly  universal 
proposition,  '  Honesty  is  the  best  policy,'  we  infer  that  '  what 
is  not  the  best  policy  is  not  honesty,'  and  also,  'what  is  not 
honesty  is  not  the  best  policy.'  From  this  proposition,  in  fact, 
we  can  draw  two  contrapositives  ;  *  but  the  reader  will  care- 
fully remember  that  from  the  ordinary  unquantified  proposi- 
tion A  we  can  only  draw  one  contrapositive.  Thus  if  '  metals 
are  elements,'  we  must  not  say  that  '  what  are  not  metals  are 
not  elements.'  But  if  we  quantify  the  predicate  thus,  '  All  the 
metals  are  some  elements,'  we  may  infer  that  'what  are  not 
metals  are  not  some  elements.'  Immediate  inference  by  added 
determinant  and  complex  conception  can  also  be  applied  in 
either  direction  to  quantified  propositions  without  fear  of  the 
errors  noticed  under  those  heads." 

The  quantification  of  the  predicate  adds  four  more  propo- 
sitions to  those  of  the  quantified  subject,  A,  E,  I,  O,  and 
Thompson  employed  new  symbols  for  them,  U  and  Y  for  the 
affirmative,  and  the  Greek  letters  y  and  m  for  the  negative.  U 
and  Y  represent  the  predicate  as  distributed  in  the  affirma- 
tive propositions,  and  y  and  w  as  undistributed  in  the  nega- 

*  Jevons  here  uses  the  term  contrapositke  in  a  sense  which  is  different 
from  the  usual  definition  of  it,  and  to  denote  two  distinct  processes.  We 
wisli  logicians  could  adopt  it  to  denote  the  inference  to  complementary 
propositions,  and  limit  the  use  of  the  term  contrarersion  to  what  is  gener- 
ally called  contraposition.  Instead  of  speaking  of  "  two  contrapositives," 
therefore,  Jevons  might  say  a  contrapositive  and  a  contraverse.  The 
conditions  are  different  for  inferring  that  "what  is  not  honesty  is  not 
the  best  policy,"  from  those  in  which  we  can  infer  that  "  what  is  not  the 
best  policy  is  not  honesty,"  except  with  a  quantified  predicate.  See  also 
Antithesis  (p.  1G9). 


20G 


ELEMENTS  OF  LOGIC 


tive.     The  following  table  represents  the  forms  of  the  eight 
propositions  : 

U     All  Sis  all  P 

A    All  S  is  some  P 

Athrmative  propositions. 


>  Negative  propositions. 


I 

Some  S  is  some  P 

Y 

Some  S  is  all  P. 

E 

No  S  is  any  P 

V 

No  S  is  some  P 

() 


Some  S  is  not  some  P 
Some  S  is  no  P. 


The  existence  of  these  eight  propositions  makes  it  neces- 
sary to  extend  the  number  of  valid  moods  from  24,  or  rather 
19,  omitting  the  weakened  conclusion,  to  108,  without  the 
fourth  Figure,  and  144  with  that  Figure.  Sir  William  Ham- 
ilton did  not  include  this  Figure  in  his  table  and  notation,  nor 
did  Thompson,  to  whose  table  we  add  the  moods  of  the  fourth 

Figure  : 

Table  of  Moods  of  the  Syllogism. 


Figure  I. 

Figure  n. 

Figure  DI. 

Figure  IV. 

Affirm- 

Neg- 

Affirm- 

Neg- 

Affirm- 

Neg- 

Affirm- 

Neg- 

ative. 

ative. 

ative. 

ative. 

ative. 

ative. 

ative. 

ative. 

I 

uuu 

EUE 
UEE 

UUU 

EUE 
UEE 

UUU 

EUE 
UEE 

UUU 

EU  E 
D  B  E 

II 

AY  I 

i)  Y  a 
AO  co 

Y  YI 

O  Y  o. 
YO  to 

A  A  I 

T)    A    0) 

A    7)     CO 

Y  A  I 

CO     A    CO 
Y     IJ     CO 

III 

AAA 

7?     A    r, 

A  r,  n 

Y  A  A 

OA, 
Y  ri  n 

AY  A 

ri  Y  r, 
AO  v 

A  AY 

0  Y  r, 
YO   r, 

IV 

Y  Y  Y 

0  YO 
YOO 

A  Y  Y 

n  YO 
AOO 

YAY 

0  A  O 

Y   r,  O 

YY  A 

r,   A  O 
A    r,0 

V 

All 

7)       I      CO 

Auu 

Y  I  I 

Y   CO    CO 

All 

q    I    co 

A   CO     CO 

Y  I  I 

0     I      CO 
Y    CO    CO 

VI 

I  Y  I 

O)    Y    O) 
I     0    0> 

I  Y  I 

CO    Y   CO 
I     O    OJ 

I  A  I 

0  A  co 

1  71      0} 

I  A  I 

0  A  co 

1  r)    co 

VII  ... . 

UY  Y 

EYO 

UOO 

UY  Y 

EYO 

UOO 

UA  Y 

E  AO 

U   r,  O 

UA  Y 

E  A  0 
U  n  O 

VIII... 

AU  A 

r,    U    r, 

AE, 

YU  A 

OU  v 
YE  „ 

AUA 

1   U    v 
AE   r, 

YU  A 

0  U  r, 
YE  n 

IX 

U  A  A 

E  AE 

U  v  n 

UA  A 

E  A  E 

U  r,  v 

U  Y  A 

EYE 

UO     7, 

U  Y  A 

EYE 

U    O     7, 

X 

Y  U  Y 

O  UO 
Y  EE 

AUY 

r,  UO 
AEE 

Y  U  Y 

OUO 
YEE 

AUY 

r,    UO 
A  EE 

XI 

U  I  I 

E  I  0 

U     CO    CO 

U  I  I 

E  I  O 

U   CO    0) 

U  I  I 

E  I  0 

U  co  co 

U  I  I 

E  I  O 

U    CO     CO 

xn  . . . . 

I  U  I 

CO    U    CO 

I    E    77 

I  U  I 

0)    U    0) 

IE» 

I  U  I 

CO    U    CO 
I    E      7, 

I  U  I 

CO     U   CO 
I    E    7? 

QUANTIFICATION  OF  THE  PREDICATE  267 

In  this  table  the  columns  marked  affirmative  and  negative 
represent  respectively  the  affirmative  and  negative  conclusions 
which  the  moods  of  the  quantified  predicate  will  give.  It 
omits  the  cases  of  weakened  conclusion.  It  is  interesting  to 
remark  that  the  negative  conclusions  are  twice  as  many  as  the 
affirmative.  The  large  number  of  both  of  them  adds  very  much 
to  the  difficulties  of  remembering  those  that  are  valid  and 
those  that  are  invalid.  It  would  seem  that  instead  of  simpli- 
fying the  process  of  reasoning  the  quantification  of  the  predi- 
cate very  much  complicates  it.  In  some  cases  it  certainly 
does  simplify  it,  but  in  so  many  other  cases  it  comj)licates  it 
that  little  is  to  be  gained  by  the  system.  But  its  theoretical 
principle  should  be  considered. 

Hamilton's  statement  of  its  value  is  brief  and  to  the  point. 
He  says  the  fact  "  that  we  can  only  rationally  deal  with  what 
we  already  understand,  determines  the  simple  logical  postu- 
late—  To  state  explicitly  what  is  thought  implicitly.  From  the 
consistent  application  of  this  postulate,  on  which  Logic  ever 
insists,  but  which  logicians  have  never  fairly  obeyed,  it  fol- 
lows, that,  logically,  we  ought  to  take  into  account  the  quan- 
tity, always  understood  in  thought,  but  usually,  and  for  mani- 
fest reasons,  elided  in  its  expression,  not  only  of  the  subject, 
but  also  of  the  predicate  of  a  judgment.  This  being  done, 
and  the  necessity  of  doing  it  will  be  proved  against  Aristotle 
and  his  repeaters,  we  obtain,  inter  alia,  the  ensuing  results : 

1.  "  That  the  preindesifjnate  *  terms  of  a  proposition,  whether 
subject  or  predicate,  are  never,  on  that  account,  thought  as 
indefinite  (or  indeterminate)  in  quantity.  The  only  indefinite 
is  particular,  as  opposed  to  definite  quantity  ;  and  this  last,  as 
it  is  either  of  an  extensive  maximum  undivided,  or  of  an  ex- 

*  Hamilton  employs  the  terms  predesignate  and  preindesignate  to  de- 
note the  two  subdivisions  of  Particular  propositions  or  terms.  "  Predesig- 
nate  "  denotes  what  we  have  called  General  terms  and  propositions,  as 
"Man  is  wise,"  where  we  may  mean  all  or  some,  the  quantity  being  in- 
differently expressed.  "  Preindesignate  "  denotes  the  ordinary  particu- 
lar proposition,  as,  "Some  men  are  wise,"  where  the  subject  is  definitely 
indefinite. 


26S  ELEMENTS  OF  LOGIC 

tensive  minimum  indivisible,  constitutes  quantity  universal 
(general)  and  quantity  singular  (individual).  In  fact,  definite 
and  indefinite  are  the  only  quantities  of  which  we  ought  to 
hear  in  Logic  ;  for  it  is  only  as  indefinite  that  particular,  it  is 
only  as  definite  that  individual  and  general,  quantities  have 
any  (and  the  same)  logical  avail. 

2.  "The  revocation  of  the  two  terms  of  a  proposition  to  their 
true  relation ;  a  proposition  being  always  an  equation  of  its 
subject  and  predicate. 

3.  "The  consequent  reduction  of  the  Conversion  of  Proposi- 
tions from  three  species  to  one  ;  that  of  Simple  Conversion. 

4.  "  The  reduction  of  all  the  General  Laws  of  Categorical 
Syllogisms  to  a  Single  Canon. 

5.  "  The  evolution  from  one  canon  of  all  the  species  and  va- 
rieties of  Syllogism. 

6.  "  The  abrogation  of  all  the  Special  Laivs  of  Syllogism. 

7.  "A  demonstration  of  the  exclusive  possibility  of  Three 
syllogistic  Figures,  and  (on  new  grounds)  the  scientific  and 
final  abolition  of  the  Fourth. 

8.  "  A  manifestation  that  Figure'  is  an  unessential  variation 
in  syllogistic  form  ;  and  the  consequent  absurdity  of  reducing 
the  syllogisms  of  the  other  figures  to  the  first. 

9.  "An  enouncement  of  one  Organic  Principle  for  each  Fig- 
ure. 

10.  "  A  determination  of  the  true  number  of  legitimate 
Moods;  with 

11.  "  Their  amplification  in  number  {thirty-six). 

12.  "  Their  numerical  equality  under  all  the  figures  ;  and 

13.  "  Their  relative  equivalence,  or  virtual  identity,  through- 
out every  schematic  difference." 

The  remaining  points  of  modification  and  advantage  are 
mainly  of  interest  to  advanced  scientific  Logic,  and  need  not 
be  repeated  here.  Even  some  of  those  we  have  quoted  are 
not  of  any  apparent  importance  to  practical  Logic.  But  all 
of  them  present  a  formidable  number  of  consequences  imputed 
to  the  quantification  of  the  predicate.  The  first  four  and  the 
eighth  are  important  simplifications  of  reasoning,  and  the  re- 


QUANTIFICATION  OF  THE  PREDICATE  2G9 

maining  are  at  least  interesting  modifications.  But  there  are 
two  or  three  difficulties  in  the  theory  which  Hamilton  seems 
not  to  have  noticed,  and  which  go  far  to  offset  all  the  advan- 
tages of  quantifying  the  predicate. 

The  first  of  these  is  an  incident  in  the  use  of  the  word  some. 
This  term  in  the  quantification  of  the  subject,  according  to  the 
Square  of  Opposition,  must  mean  some,  and  it  may  or  it  may 
not  be  all,  and  must  not  mean  some  and  only  some.  If  it  meant 
the  latter,  the  assertion  of  I  would  always  prevent  the  asser- 
tion of  A.  Hence,  for  the  purposes  of  Logic  it  must  take 
the  former  meaning,  so  that  A  may  remain  indeterminate  and 
possible  upon  the  assertion  of  I.  But  in  the  quantification 
of  the  predicate,  if  some  mean  an  indefinite  part,  and  it  may  or 
not  be  all  of  the  predicate,  the  first  thing  to  be  remarked 
is  that  the  predicate  in  reality  remains  as  undistributed  as 
before,  and  the  second  is,  that  the  antitheses  or  complemen- 
taries  of  such  propositions  are  no  more  possible  than  be- 
fore, a  j)ossibility  which  was  said  to  be  consequent  upon 
the  quantification  of  the  predicate.  On  the  other  hand,  if 
we  use  some  to  mean  a  part,  and  only  a  part,  we  obtain  the 
quantification  of  the  predicate  in  affirmative  propositions  at 
the  expense  of  the  regular  rule  about  its  use  in  the  quan- 
tification of  the  subject ;  and  in  negative  propositions  a  se- 
rious ambiguity  arises  which  it  is  hard  to  overcome.  Thus 
propositions  n  and  w  may  each  have  two  meanings.  "No 
S  is  some  P,"  may  mean  either  that  S  is  not  any  por- 
tion of  P,  or  that  it  is  not  a  certain  indefinite  portion  of  P. 
So  in  the  proposition  "  Some  S  is  not  some  P."  This  am- 
biguity appears  most  distinctly  when  we  come  to  draw  the 
antitheses  or  complementaries  of  such  propositions.  Thus 
in  proposition  U,  "  All  men  are  all  rational,"  we  can  imme- 
diately infer,  as  in  exclusive  propositions,  "  All  not-men  are 
not  rational."  "When  the  use  of  some  in  the  predicate  means 
a  part,  and  only  a  part,  the  complementary  can  be  inferred 
that  all  which  is  not  the  subject  is  not  that  portion  of  the 
predicate.  But  the  terms  indefinitely  considered  would  imply 
that  it  was  not  the  other  portion   also.     Thus,   "  All  metals 


270  ELEMENTS  OF  LOGIC 

are  some  elements,"  ought  to  give  "  All  not-metals  are  not 
some  elements,"  and  so  it  will,  if  we  mean  the  particular  some 
identified  with  the  subject.  But  the  complementary  may 
mean  not  any  "  some  "  of  the  elements,  which  cannot  possibly 
be  true.  These  ambiguities  are  a  serious  difficulty  in  the  way 
of  the  Hamiltonian  theory. 

There  is  a  second  important  difficulty  which  shows  how  lit- 
tle the  accepted  rules  of  syllogistic  reasoning  can  be  relied 
upon  to  determine  the  validity  of  the  new  cases  introduced 
by  the  quantification  of  the  predicate.  It  is  remarked  by 
Keynes.  In  ordinary  reasoning  the  distribution  of  the  middle 
term  in  at  least  one  of  the  premises,  and  the  retention  in  the 
conclusion  of  the  same  quantity  of  the  major  and  minor  terms 
as  in  the  premises,  are  a  guarantee  of  valid  inference.  But  in 
the  mood  AUA  of  a  quantified  predicate  and  of  the  second 
Figure,  the  observance  of  these  conditions  is  no  security 
against  a  fallacy,  as  will  be  apparent  in  the  following  syllogism 
and  its  symbolic  representation  in  Fig.  26. 

All  P  is  some  M 
All  S  is  all  M 
.• .  All  S  is  all  P  or  some  P. 

The  circles  will  bring  out  the  fallacy  more  clearly. 


Fig.  26. 

It  is  evident  from  this  representation  that  we  cannot  infer 
from  the  premises  that  all  S  is  either  all  or  some  P.  All  that 
we  could  infer  is  that  "Some  S  is  all  P,"  which  is  Y  or  "  Some 
S  is  some  P,  which  is  w ;"  and  yet  neither  illicit  middle,  nor 
illicit  major,  nor  illicit  minor,  is  committed  in  the  propositions 
which,  it  seems,  ought  to  follow,  namely,  "All  S  is  all  P,"  etc.  ; 
as  a  consequence  of  this,  Keynes  finds  it  necessary  to  add  the 


QUANTIFICATION  OF  THE  PREDICATE  271 

following  new  rule  to  the  list  already  given  for  determining 
valid  syllogisms  :  If  one  premise  is  U,  while  in  the  other  premise 
the  middle  term  is  undistributed,  then  the  term  combined  with 
the  middle  term  in  the  U  premise  must  be  undistributed  in  the 
conclusion. 

But  this  addition  of  a  rule  only  complicates,  instead  of  sim- 
plifies, the  process  of  reasoning  as  we  desire  to  use  it  in  prac- 
tical life  ;  and  the  immense  number  of  valid  moods  has  the 
same  effect,  while  the  ordinary  rules  of  reduction  to  the  first 
Figure,  and  the  common  practice  of  reasoning  in  this  Figure 
makes  the  matter  as  simple  as  it  can  be  made,  taking  into  ac- 
count the  nature  of  language  and  the  defects  of  expression  in 
our  thought.  These,  then,  with  other  incidents,  are  defects 
in  the  practical  importance  of  the  quantification  of  the  predi- 
cate. 

Nevertheless  there  is  one  consideration  of  great  interest 
and  importance  in  it,  which,  if  we  cannot  make  as  much  prac- 
tical use  of  it  as  might  be  desirable,  does  explain  the  simpli- 
city of  mathematical  reasoning — the  use  of  exclusive  proposi- 
tions and  definitions,  and  the  habitual  tendency  of  ordinary 
minds  to  use  predicates  as  if  they  were  definitely  quantified. 
Hamilton's  dictum  is  a  correct  one,  namely,  that  ready  rea- 
soning is  greatly  facilitated,  and  rendered  less  liable  to  fallacy 
by  stating  explicitly  what  is  implicitly  thought.  This  means 
that  we  should  state  definitely  what  is  definitely  thought,  and 
not  state  it  indefinitely.  If  I  mean  that  "  All  men  are  wise," 
"  All  governments  are  good,"  I  should  say  so,  and  not  conceal 
a  subterfuge  under  the  indefinite  "Man"  or  "Government," 
etc.  But  in  such  propositions  the  predicate  remains  as  in- 
definite in  extent  or  quantity  as  the  subjects  just  mentioned. 
If  only  I  knew  in  such  statements  whether  the  subject  was  the 
whole  or  the  part  of  the  predicate,  I  could  know  better  how 
to  use  them.  Thus  if  I  mean  by  the  first  proposition  that 
"All  men  are  all  the  wise,"  I  know  first  that  "All  who  are 
not-men  are  not  wise,"  and  second,  that  the  two  terms  are 
perfectly  convertible.  This  is  precisely  what  occurs  in  ex- 
clusive propositions  and  definitions.     In  the  former  class  the 


272  ELEMENTS  OF  LOGIC 

predicate  is  definitely,  and,  we  might  say,  formally,  quantified 
by  the  use  of  only  qualifying  the  subject,  and  which  indicates 
that  the  quantity  of  the  predicate  is  not  greater  than  that  of 
the  subject.  In  definitions  the  predicate,  although  not  for- 
mally, is  materially  quantified,  in  being  made  identical  with 
the  subject,  and  so  convertible  with  it.  The  effect  of  such 
propositions  upon  the  reasoning  is  to  make  it  valid  in  some 
instances  where  there  appears  to  be  a  distinct  formal  fallacy. 
Take  an  illustration  with  an  exclusive  proposition  : 

Only  virtue  is  praiseworthy. 
Courage  is  praiseworthy. 
Therefore  courage  is  a  virtue. 

This  appears  from  the  form  of  the  propositions  to  be  a  syllo- 
gism in  AAA  of  the  second  Figure,  and  therefore  invalid  be- 
cause of  the  undistributed  middle,  and  yet  in  spite  of  this  we 
cannot  resist  the  impression  that  the  inference  is  correct. 
The  reason  for  this  is  that  the  virtual  quantification  of  the 
predicate  in  the  major  premise  totally  modifies  the  reasoning 
as  compared  with  the  ordinary  rules.  We  can  therefore  ex- 
plain the  case  in  two  ways.  First,  since  the  subject  and 
predicate  are  made  coextensive  by  the  use  of  only,  the  major 
premise  is  equivalent  in  meaning  to  its  converse,  namely, 
"  What  is  praiseworthy  is  virtue,"  and  this  change  turns  the 
syllogism,  in  spite  of  its  present  form,  into  AAA  of  the  first 
Figure,  which  is  valid.  We  may  conceive  that  this  is  the  or- 
der of  thought,  in  spite  of  the  order  of  expression,  and  the 
right  to  so  conceive  it  depends  wholly  upon  the  definitely 
quantified  predicate.  Second,  since  the  quantification  of  the 
predicate  in  the  major  premise,  making  it  coextensive  with 
the  quantity  of  the  subject,  has  the  effect  of  distributing  it, 
we  have  a  syllogism  of  the  second  Figure  which  distributes 
the  middle  term  at  least  once  in  the  premises,  in  spite  of  the 
affirmative  character  of  the  propositions.  Tliis  distribution 
enables  us  to  draw  the  inference  which  appears  in  the  con- 
clusion. In  the  case  of  definitions  the  illustration  would  be 
the  same. 


QUANTIFICATION  OF  THE  PREDICATE  273 

Common  reasoning  often  treats  propositions  as  if  they 
were  either  definitions  or  exclusive  propositions,  and  it  is  be- 
cause the  thought  of  the  person  reasoning  is  conceived  as 
representing  a  certain  identity  between  subject  and  predicate 
which  may  not  be  expressed,  but  is  in  mind.  Hence  the  dis- 
position to  substitute  them  for  each  other.  Now  this  substi- 
tution can  be  done  with  impunity  whenever  the  predicate  is 
made  quantitatively  equal  to  the  subject,  and  so  the  need  of 
distinguishing  between  propositions  which  intend  to  dis- 
tribute and  those  which  do  not  intend  to  distribute  the  predi- 
cate. If  common  usage  could  adopt  a  symbol  which  could 
serve  this  purpose  many  logical  difficulties  would  be  over- 
come. The  theory  of  pure  Logic  would  thus  find  its  condi- 
tions supplied  in  practice.  But  as  it  is,  all  propositions,  ex- 
cept definitive,  exclusive,  and  mathematical  propositions,  are 
alike  in  their  form,  while  some  of  them  in  their  matter,  or  the 
way  in  which  the  mind  thinks  them,  may  be  quantified,  so  that 
the  syllogisms  containing  them  cannot  be  judged  by  the  for- 
mal laws  of  reasoning  until  we  know  what  content  they  are  sup- 
posed to  have.  Thus  we  may  say,  "  Houses  are  residences,"  and 
formally  the  predicate  is  not  quantified,  or  is  not  distributed, 
and  cannot  be  used  with  an  affirmative  minor  premise  in  the 
second  Figure.  But  if  we  have  in  thought  the  notion  that  the 
predicate  is  identical  with  the  subject  it  is  quantified,  and  the 
reasoning  is  altered.  This,  then,  in  brief,  is  the  advantage 
of  quantifying  the  predicate.  It  indicates  explicitly  whether 
all  or  only  a  part  of  it  is  taken  into  account. 

It  must  be  observed,  however,  that  in  propositions  I  and  Y 
no  special  advantage  is  gained,  because  they  leave  the  predi- 
cate undistributed,  except  that  we  know  definitely  from  their 
form  how  to  understand  them.  The  chief  importance  attaches 
to  the  distinction  between  A  and  U,  which,  if  the  quantifica- 
tion were  practicable  in  most  cases,  would  enable  us  to  know 
the  extent  of  the  predicate  as  well  as  that  of  the  subject.  But 
even  if  not  of  much  service  practically,  the  theory  shows  veiy 
clearly  what  can  be  done  with  definite  propositions,  and  how 
the  common  mind  often  acts  in  its  reasoning,  when  the  trained 
18 


274  ELEMENTS  OF  LOGIC 

logician  would  impute  a  violation  of  the  formal  laws  of  Logic 
to  it.  Besides,  it  explains  a  common  tendency  of  the  mind 
to  carry  on  the  substitution  of  one  term  for  another  without 
stopping  to  consider  their  distribution.  The  substitution  in 
an  unqualified  manner  is,  no  doubt,  an  error,  but  it  is  rather 
an  error  of  assumption  in  many  cases  than  of  inference.* 

*  References  on  the  doctrine  of  the  Quantification  of  the  Predicate  are 
as  follows :  Hamilton :  Lectures  on  Logic,  Appendix  V. ;  De  Morgan  : 
Formal  Logic;  Thompson:  Laws  of  Thought,  Part  II.,  Sections  77-79; 
Mill :  Examination  of  Sir  William  Hamilton's  Philosophy,  Chap.  XXII. 


CHAPTER  XX. 

MATHEMATICAL   AND    OTHER   REASONING 

The  principles  involved  in  the  quantification  of  the  predi- 
cate, and  for  that  matter,  also  of  the  subject,  will  help  us  to 
understand  the  certitude  and  exemption  from  error  displayed 
in  mathematical  reasoning,  and  also  the  difference  between 
this  and  other  reasoning.  We  shall  define  it  more  carefully 
after  illustrating  it  and  pointing  out  its  relation  to  the  princi- 
ples discussed  in  the  preceding  chapter. 

The  first  important  observation  to  be  made  is  that  in  mathe- 
matical reasoning  all  propositions  are  U  or  E propositions,  or 
their  equivalents ;  that  is,  are  universally  and  definitely  quanti- 
fied in  their  subject  and  predicate.  The  effect  of  this  fact  is  that 
which  was  remarked  by  Hamilton  in  the  eighth  observation 
we  quoted  from  him  (p.  268).  It  makes  subject  and  predi- 
cate quantitatively  equal,  so  that  we  can  dispense  with  the 
variations  of  Figure  in  Syllogisms,  in  so  far  as  the  validity  of 
the  reasoning  is  concerned.  All  the  Figures  are  valid  alike, 
and  from  what  has  been  said  about  the  nature  of  the  propo- 
sitions in  mathematical  reasoning  the  Moods  will  be  only 
three,  namely,  UUU,  UEE,  and  EUE,  all  of  which  are  valid  in 
all  the  Figures.  In  the  following  illustrations  the  sign  of 
equality  serves  both  as  the  copula  and  the  sign  of  equivalence, 
and  the  negative  symbol  previously  explained  (p.  140)  denotes 
inequality  as  well  as  negation.  The  letters  denote  the  various 
terms  as  before,  only  they  stand  for  numbers  or  quantities. 
Mathematical  reasoning,  then,  will  appear  in  the  following 
forms  indifferently  : 


1st  Fig. 

2d  Fig. 

3d  Fig. 

4th  Fig. 

UUU 

UUU 

UUU 

UUU 

M  =  P 

P  =  M 

M  =  P 

P  =  M 

S  =M 

S  =M 

M  =  S 

M=S 

.S  =  P 

,-.  S  =P 

.-.  S  =P 

.  • .  S  =  P. 

276  ELEMENTS  OF  LOGIC 


1st  Fig. 

2d  Fig. 

3d  Fig. 

4th  Fig. 

UEE 

UEE 

UEE 

UEE 

M=  P 

P  =  M 

M  =  P 

P  =  M 

S   X  M 

S  X  M 

M  x  S 

MX  S 

s  x  p    . 

•.SxP 

.-.SxP     . 

,  • .  S  X  P. 

If  we  turn  to  the  table  of  Moods  with  the  quantified  predicate 
we  shall  find  that  these  two  are  valid  in  all  four  Figures,  a 
fact  which  is  due  to  understanding  definitely  that  the  predi- 
cate is  limited  to  the  subject,  or  made  equal  to  it  in  exten- 
sion. The  process  for  all  four  Figures  can  be  represented 
symbolically  by  the  following  circles  : 


Fio.  27.  Fie.  28. 

Fig.  27  represents  the  Mood  UUU,  and  Fig.  28  the  Mood 
UEE.  Mood  EUE  would  be  the  same  as  Fig.  28,  only  the 
letters  S  and  P  would  change  jriaces.  It  is  very  evident  from 
this  how  simple  mathematical  reasoning  is,  and  how  little  re- 
gard we  need  to  pay  to  Figure  in  the  form  of  presenting  it. 
If  all  reasoning  were  the  same,  it  would  be  less  exposed  to 
formal  fallacies,  and  undoubtedly  it  is  the  tendency  of  com- 
mon minds  to  construe  their  reasoning  mathematically,  that 
explains  both  their  liability  to  formal  fallacies  and  the  neces- 
sity of  being  on  our  guard  for  the  existence  of  material  fal- 
lacies in  it  when  it  is  formally  correct.  The  untrained  mind, 
accustomed  to  deal  with  its  terms  and  their  import  in  a 
mathematical  sense,  very  easily  confuses  with  them  other 
terms  and  propositions  having  the  same  form  as  the  mathe- 
matical, and  is  at  a  loss  to  discover  its  liability  to  error. 

The  illustrations  and  their  symbolic  representation  enable 
us  to  define  mathematical  reasoning  and  to  distinguish  it  from 
the  species  which  might  be  called  logical,  but  for  the  fact  that 
all  forms  of  reasoning  are  logical.     It  could,  however,  be  called 


MATHEMATICAL  AND   OTHER  REASONING        277 

"logical"  in  the  sense  in  which  we  have  distinguished  be- 
tween the  "mathematical"  and  the  "logical"  genus.  But 
without  pressing  this  meaning,  mathematical  reasoning  is 
that  which  deals  •purely  with  quantity.  It  maybe  called  pure 
quantitative  reasoning,  in  distinction  from  the  second  form, 
which  may  be  called  mixed  qualitative  and  quantitative.  Math- 
ematical reasoning  is  based  upon  relations  of  quantity  which 
expresses  no  variations  or  differences  except  in  amount ; 
quanto-qualitative  reasoning  is  not  only  based  upon  rela- 
tions of  quality  but  of  quantity  also,  and  in  addition  to 
differences  of  degree  or  amount,  represents  differences  of 
kind.  This  distinction  between  the  two  processes  merely 
means  that  the  conceptions  employed  in  mathematical  rea- 
soning either  deal  exchisively  with  the  abstract  ideas  of 
quantity,  or  deal  with  the  concrete  objects,  only  in  numerical 
terms  of  time,  space,  force,  etc.,  or  commensurable  quantities. 
They  are  not  viewed  as  a  group  of  qualities,  but  as  individual 
wholes.  When  we  say  "  all  men,"  "all  animals,"  "  all  citizens," 
we  speak  of  a  number  of  individual  persons  as  individuals,  and 
do  not  take  into  account  their  various  differences.  They  may 
be  of  different  sizes,  different  color,  or  of  different  powers, 
but  the  "  all "  does  nothing  but  describe  them  numerically. 
When  the  predicate  is  thus  quantified,  the  identity  between 
subject  and  predicate  becomes  purely  a  quantitative  identity, 
and  so  the  terms  are  equated.  But  when  we  say  "man,"  " an- 
imal," "citizen,"  etc.,  we  think  of  attribute  wholes,  so  that  any 
identity  between  them  and  a  predicate  is  qualitative,  and  in  my 
judgment  is  not  of  the  nature  of  an  equation  at  all.  Its  real 
import  may  be  expressed  in  one  of  two  ways,  according  as  the 
predicate  is  an  attribute  or  a  class  term.  We  have  already 
spoken  of  the  two  kinds  of  judgment,  the  intensive  and  the 
extensive  (p.  123).  The  former,  as  for  example,  "  Man  is 
wise,"  expresses  by  its  predicate  that  a  certain  group  of  quali- 
ties called  man  is  accompanied  by  the  quality  wise,  or  that  amid 
that  group  of  qualities  will  be  found  one  denoted  by  the  term 
"  wise."  The  so-called  "  identity,"  or  "  agreement "  between 
the  subject  and  predicate  in  such  cases,  is  not  one  of  quan- 


278  ELEMENTS  OF  LOGIC 

tity.  The  relation  should  rather  be  called  that  of  connection,  a 
relation  that  cannot  be  dealt  with  quantitatively,  except  as  it 
may  be  common  or  invariable.  The  extensive  judgment,  as 
"  Man  is  an  animal,"  expresses  by  its  predicate  a  distinct  rela- 
tion of  identity  between  the  subject  and  predicate,  namely,  that 
the  two  have  certain  common  qualities  by  which  they  may  be 
classified  together.  But  no  relation  of  quantity  appears  in 
this  case  any  more  than  in  the  intensive  judgment.  In  both, 
however,  there  is  nothing  said  to  prevent  any  number  of  other 
and  different  beings  from  being  connected  with,  or  included 
in,  the  predicate.  The  relation,  therefore,  is  one  which  is 
based  upon  the  real  or  the  possible  coexistence  of  resem- 
blances and  differences  between  certain  objects  connected  as 
subject  and  predicate.  This  is  only  to  say  that  in  what  we 
have  called  qualitative  judgments  and  reasoning  the  concep- 
tions with  which  we  have  to  deal  are  genera  and  species,  es- 
sentia and  accidentia,  or  conferentia  and  differentia.  These  con- 
ceptions do  not  appear  as  such  in  mathematical  reasoning. 
They  prevent  that  definite  quantification,  especially  of  the 
predicate,  which  is  necessary  to  make  the  reasoning  mathe- 
matical. 

But  there  is  a  class  of  judgments,  intensive  or  extensive, 
just  as  we  choose  to  regard  them,  in  which  the  identity  be- 
tween subject  and  predicate  is  such  as  to  preclude  the  admis- 
sion of  any  other  than  the  existing  subject  in  the  same  con- 
nection. They  are  such  judgments  as  "  Virtue  is  goodness," 
"Quadrupeds  are  four-footed  animals,"  "Honesty  is  the  best 
policy,"  "  Government  is  social  organization,"  etc.  Here  there 
is  a  qualitative  connection,  or  identity  of  qualities,  expressed 
by  the  propositions.  But  it  is  an  absolute  identity.  The 
terms  are  either  synonymous  or  represent  definitions.  The 
relation  expressed  is  not  that  between  genus  and  species, 
or  the  connection  between  essentia  and  accidentia,  or  con- 
ferentia and  differentia,  but  between  genus  and  genus,  es- 
sentia and  essentia,  conferentia  and  conferentia,  accidentia 
and  accidentia,  or  differentia  and  differentia.  But  such  a 
relation  coincides  exactly  with  the  quantitative  relation,  and, 


MATHEMATICAL  AND   OTHER  REASONING       279 

so  far  as  reasoning  is  concerned,  can  be  made  convertible 
with  it.  Hence,  in  all  such  cases  the  reasoning  with,  such 
propositions  is  as  assured  as  that  in  mathematics,  because 
their  quantitative  meaning  coincides  with  their  qualitative, 
or  their  qualitative  with  their  quantitative,  and  so  can  be 
substituted  in  its  stead.  It  will  be  self-evident  from  this  fact 
that  the  whole  process  might  be  greatly  simplified  if  ah1  terms 
could  be  made  to  represent  a  similar  connection  between  the 
subject  and  predicate. 

The  peculiarity  of  the  relation  between  the  two  terms  in 
these  two  classes  of  propositions  and  in  all  mathematical  judg- 
ments might  justify  the  use  of  the  name  Traduction  for  this 
species  of  reasoning,  in  contrast  with  ordinary  Deduction  on 
the  one  hand,  and  Induction  on  the  other.  It  does  not  differ 
essentially,  however,  from  Deduction,  as  the  principle  is  the 
same.  Derived  from  trans  and  duco,  to  lead  over,  it  might 
denote  the  substitution  which  is  characteristic  of  all  reason- 
ing that  involves  a  predicate  in  its  propositions  quantitatively 
identical  with  the  subject.  Jevons  alludes  to  the  use  of  this 
term  in  connection  with  a  species  of  syllogistic  form  and 
reasoning  which  we  have  yet  to  consider.  But  he  does  not 
employ  it  to  denominate  the  essential  characteristic  of  mathe- 
matical reasoning,  which  can  carry  on  its  substitution  without 
regard  to  the  question  of  Figure,  as  we  have  already  seen. 
All  its  propositions  are  definite,  and  so  are  its  terms.  Even 
those  conceptions  which  denote  a  part  of  a  whole  are  definite. 
Mathematical  conceptions  are  not  qualified  by  the  indefinite 
some,  which  may  denote  any  portion  whatever  less  than  the 
whole,  but  they  are  expressed  by  some  definite  fraction  of 
the  whole,  which  is  equivalent  to  a  universal  notion  in  its  im- 
port, whenever  a  portion  of  some  larger  total  is  to  be  reckoned 
with.  Even  the  unknown  quantities  of  Algebra  are  no  ex- 
ception to  this  principle.  They  always  represent  definite 
quantities,  which  are  called  "  unknown  "  because  they  may  be 
used  for  any  fixed  number  we  please.  In  all  instances  of  such 
terms,  however,  having  a  mathematical  import,  we  have  for 
our  propositions  an  equation  or  equations  in  which  the  pro- 


280  ELEMENTS  OF  LOGIC 

cess  of  transition  from  one  term  to  another  is  substitution  or 
traduction.  Mathematical  reasoning  thus  embodies  in  a  per- 
fect form  the  principles  most  clearly  enunciated  in  the  doc- 
trine of  the  quantification  of  the  predicate,  although  implied 
in  the  quantification  of  the  subject.  It  is  simply  the  substi- 
tution of  one  term  for  another  which  involves  it,  because  they 
are  quantitatively  identical  at  the  same  time  that  they  may 
be  necessarily  connected  in  another  way. 

The  principle  thus  established  can  be  used  to  explain  a 
form  of  reasoning  which  Jevons  regards  as  irregular,  and  yet 
valid  in  spite  of  its  real  or  apparent  violation  of  the  formal 
laws  of  the  syllogism.  The  first  illustration  chosen  by  him  is 
not  formally  incorrect,  but  is  regarded  by  him  as  irregular. 
It  is  as  follows  : 

The  sun  is  a  thing  insensible. 

The  Persians  worship  the  sun. 

Therefore  the  Persians  worship  a  thing  insensible. 

The  only  apparent  irregularity  in  this  example  is  the  use  of  a 
part  of  the  predicate  of  the  minor  premise  in  the  conclusion, 
and  the  consideration  of  the  other  part  of  it  as  the  middle 
term.  But  logically  I  can  see  no  objection  to  this  process,  nor 
a  reason  in  it  for  regarding  the  syllogism  as  in  any  way  ir- 
regular. It  is  true  that,  grammatically  considered,  the  subject 
of  the  major  premise  is  "  the  sun,"  and  the  predicate  of  the 
minor  premise  is  "  worship  the  sun,"  so  that  being  AAA  of 
the  first  Figure,  they  occupy  the  place  of  the  middle  term. 
The  difference  between  them  might  seem  to  make  a  fourth 
term,  and  therefore  a  fallacy  of  Quaternio  Terminorum.  This 
would  undoubtedly  be  the  case  if  we  attempted  to  draw  tbe 
conclusion,  "  The  Persians  are  a  thing  insensible."  But  the 
whole  matter  is  altered  when  we  transfer  the  term  "  worship  " 
to  the  conclusion,  because  it  signifies  that  the  real  middle 
term  of  thought  is  "the  sun,"  and  the  reasoning  becomes  per- 
fectly valid,  and  is  no  exception  to  the  ordinary  form  of  syllo- 
gistic inference.     But  it  is  not  a  case  of  mathematical  substi- 


MATHEMATICAL  AND   OTHER  REASONING        281 

tution,  except  as  a  part  of  the  apparent  middle  term  is  trans- 
ferred to  the  conclusion. 

The  next  instance  is  somewhat  similar,  and  yet  is  as  dif- 
ferent in  other  respects.  Jevons  is  correct  when  he  asserts 
that  a  great  deal  of  our  reasoning  is  done  after  this  manner. 
The  following  is  his  illustration  : 

The  Divine  Law  commands  us  to  honor  kings. 

Louis  XIV.  is  a  king. 

Therefore  the  Divine  Law  commands  us  to  honor  Louis  XTV. 

The  peculiarity  of  this  instance  is  that  it  seems  to  be  a  case  of 
AAA  in  the  second  Figure,  namely,  an  example  of  illicit  mid- 
dle, and  yet  valid  in  spite  of  the  fact.  As  before,  we  substi- 
tute in  the  conclusion  a  part  of  the  predicate  in  the  minor 
premise,  and  this  leaves  the  term  "king"  as  the  middle  term. 
Why,  then,  does  the  reasoning  impress  us  irresistibly  with  its 
validity,  although  the  form  is  against  this  feeling  ?  If  we 
stated  the  case  as  follows  there  would  be  no  difficulty  in  re- 
jecting the  conclusion  : 

Louis  XTV.  commands  us  to  honor  kings. 
The  Divine  Law  commands  us  to  honor  kings. 
Therefore  the  Divine  Law  is  Louis  XD7. 

The  fallacy  is  palpable  in  this  instance.  Why  is  it  not  so  in 
the  former? 

Two  explanations  can  be  given  to  the  case  which  will  an- 
swer the  question.  The  first  is  a  reduction  of  the  thought 
expressed  to  the  proper  Mood  and  Figure.  Thus  the  minor 
premise  of  the  original  example  can  be  interpreted  as  mean- 
ing "  Kings  are  those  whom  the  Divine  Law  commands  us  to 
honor."  This  is  precisely  the  same  thought  as  before,  and  it 
gives  AAA  of  the  first  Figure,  with  the  conclusion,  "  Louis 
XIV.  is  he  whom  the  Divine  Law  commands  us  to  honor." 
In  the  premise  and  conclusion,  therefore,  we  have  simply  the 
passive  for  the  active  form  of  expression,  while  the  logical 
import  is  precisely  the  same  as  before.     In  this  way  we  an- 


282  ELEMENTS  OF  LOGIC 

swer  our  question  by  showing  that  logically  the  apparently 
invalid  syllogism  may  really  be  valid  by  being  of  an  actually 
different  form  in  thought  from  what  it  is  in  expression. 

But  there  is  a  second  way  of  dealing  with  the  case.  In  the 
proposition  "The  Divine  Law  commands  us  to  honor  kings,'' 
it  is  evident  that  the  predicate  is  not  distributed,  namely,  that 
other  things  might  command  the  same.  But  in  such  state- 
ments we  usually  quantify  or  universalize  certain  terms  help- 
ing to  constitute  the  predicate.  Thus  in  making  this  asser- 
tion we  are  most  likely  to  mean  all  kings.  Now,  since  a  part 
of  it  is  substituted  in  the  conclusion,  this  quantification  of 
the  term  "  king  "  has  the  effect  of  distributing  it,  that  is,  of 
indicating  that  Louis  XIV.  is  included  in  the  general  class  as 
soon  as  he  is  affirmed  to  be  a  king.  The  reasoning,  there- 
fore, becomes  a  simple  case  of  substitution  of  Louis  XTV.  for 
"  king,"  which  is  already  included  by  implication  or  asser- 
tion in  the  whole  class.  If  we  had  distinctly  said  or  thought 
that  "  the  Divine  Law  commands  us  to  honor  some  kings," 
and  then  asserted  that  "  Louis  XD7.  is  a  king,"  we  should 
have  had  no  inclination  to  draw  the  conclusion  that  Louis 
XD7.  is  to  be  honored,  and  would  have  very  quickly  j>erceived 
the  fallacy  of  attempting  to  do  so,  siniply  for  the  reason  that 
it  would  appear  to  be  a  case  of  undistributed  middle. 

But  it  remains  to  show  whether  the  case  can  be  resolved  by 
the  regular  laws  of  the  syllogism.  If  we  regard  the  univer- 
salizing or  distribution  of  the  term  "  king  "  as  bringing  the 
case  under  the  quantification  of  the  predicate,  we  should  have, 
or  should  seem  to  have,  an  instance  of  AUA  in  the  second 
Figure,  which  we  have  found  to  be  formally  invalid,  in  spite 
of  the  distribution  of  the  middle  term.  But  it  can  be  shown, 
in  spite  of  its  appearance,  that  the  reasoning  is,  after  all,  not  of 
the  second  Figure.  Thus  if  the  minor  premise  were  negative, 
"Louis  XIV.  is  not  a  king,"  we  should  have,  according  to  our 
supposition,  EUE,  which  is  valid  according  to  the  table.  But 
it  requires  only  a  glance  at  the  conclusion,  that  "  The  Divine 
Law  does  not  command  us  to  honor  Louis  XD7.,"  to  see  that 
it  is  fallacious,  because,  so  far  as  the  assertion  is  concerned,  the 


MATHEMATICAL  AND   OTIIER  REASONING        283 

Divine  Law  may  command  us  to  honor  all  other  persons  as 
well  as  kings.  If  we  examine  the  fallacy  carefully  we  shall 
find  that  it  is  the  same  as  AEE  of  the  first  Figure.  Let  it 
be  represented  by  the  following  diagram :  We  observe  in 
this  representation  that  "  kings  "  are  included 
in  those  whom  the  Divine  Law  commands  us 
to  honor,  and  also  that  Louis  XD7.  is  so  con- 
tained, but  excluded  from  the  class  "kings." 
The  representation  is  thus  precisely  like  that 

which  symbolizes  the  fallacy  of  AEE  in  the 

Fig.  29. 
first  Figure.     The  placing  of  "  Divine  Law  "  in 

a  circle  is  only  to  indicate  that  the  entire  predicate  is  not  dis- 
tributed. For  considering  the  fallacy  we  require  only  to  take 
the  other  circles  into  account.  But  the  fallacy,  being  shown 
to  be  equivalent  to  that  of  AEE  in  the  first  Figure,  creates 
the  suspicion  that  the  original  syllogism,  with  the  affirmative 
premise,  is  really  AAA  of  the  first  Figure,  although  it  appears 
to  be  more  nearly  related  to  the  second  Figure.  The  next 
diagram  will  illustrate  this  supposition.  In  this  representa- 
tion the  relation  between  the  terms  is  that  of 
the  first  Figure,  and  it  also  indicates  what  is 
implied  in  the  proposition  that  "  the  Divine 
Law  commands  us  to  honor  kings,"  namely, 
that  it  may  command  us  to  honor  others  also. 
But  it  brings  out  most  distinctly  what  is  in- 
volved in  the  quantification  of  a  term  and  the 
relation  implied  by  it  between  this  term  and  some  other  con- 
ception in  the  argument,  namely,  that  a  substitution  can  take 
place  in  any  particular  instance  included  under  a  class.  It  is 
done  in  this  case  without  regard  to  the  apparent  Figure  of  the 
Syllogism,  and  in  so  far  resembles  the  process  of  mathemati- 
cal or  quantitative  Logic.  But  the  reasoning  is  neither  a  case 
of  traduction  nor  one  of  the  ordinary  syllogistic  form  as  it  is 
expressed.  It  is  what  older  logicians  called  the  complex  syl- 
logism, and  it  has  generally  been  maintained  that  it  could  not 
be  reduced  to  the  regular  form.  But  this  we  shall  show  can 
be  done,  and  the  process  may  bring  out  more  clearly  the  dif- 


284  ELEMENTS  OF  LOGIC 

ference  between  mathematical  and  ordinary  qualitative  rea- 
soning. 

We  have  shown  that  the  reasoning  cannot  be  of  the  second 
Figure,  although  it  appears  to  be.  It  remains  to  show  that  it  is 
of  the  first,  and  so  to  confirm  the  suspicion  created  by  the  nat- 
ure of  the  fallacy  involved  in  supposing  it  to  be  of  the  second 
figure.  The  two  diagrams  show  the  complexity  of  the  rela- 
tions to  be  dealt  with.  In  the  first  place,  under  the  ordinary 
interpretation  of  the  proposition  "  Divine  Law  "  is  the  subject, 
and  is  distributed,  and  "  commands  us  to  honor  kings  "  is  the 
predicate,  and  undistributed.  This  relation  is  represented  by 
the  largest  circle  and  the  small  one  containing  "  Divine  Law," 
and  merely  denotes  that  other  sources  of  the  same  command 
might  exist,  in  so  far  as  the  proposition  says  nothing  to  the 
contrary.  But  the  predicate  is  a  complex  one,  and  it  is  not 
upon  the  whole  of  it  that  the  reasoning  depends.  This  is  evi- 
dent from  the  substitution  of  apart  of  it  in  the  conclusion.  The 
part  which  is  not  found  in  the  conclusion,  then,  is  the  real 
middle  term.  This  is  the  word  "  kings."  "  Commands  us  to 
honor,"  is  not  a  part  of  the  middle  term,  but  of  the  minor  or 
major  term,  as  the  case  may  be.  "Kings  "  is  the  term  upon 
which  the  reasoning  turns,  and  as  the  circles  represent  it,  is  to 
be  included  in  the  larger,  because  other  persons  may  be  in- 
cluded under  the  same  command.  This  relation  is  not  ex- 
cluded by  the  universal  quantification  of  the  word,  so  that  it 
cannot  be  conceived  as  a  predicate,  because  its  universal 
quantification  in  that  case  would  imply  its  coextension  with 
the  subject ;  which  would  imply  that  the  divine  law  could 
not  command  us  to  honor  any  one  else.  But  since  the 
proposition  does  not  exclude  honor  to  others  besides  "  kings," 
and  "  kings  "  is  the  middle  term  and  distributed,  the  only  pos- 
sible way  of  representing  the  fact  is  to  conceive  it  as  subject. 
But  to  conceive  it  as  subject  makes  the  proposition  the  major 
premise,  and  the  other  proposition,  "  Louis  XD7.  is  a  king," 
the  minor  premise,  and  the  form  becomes  AAA  of  the  first 
Figure,  as  we  have  already  asserted  it  must  be,  in  order  to 
make  the  reasoning:  valid. 


MATHEMATICAL  AND   OTHER  REASONING        2S5 

But  in  order  to  effect  this  transformation  a  change  in  the 
order  of  the  statement  is  required.  Since  the  object  of  thought 
is  mainly  the  middle  term,  as  in  such  cases  it  always  is,  there 
is  no  objection  to  the  process  which  explains  how  the  mind 
really  treats  it,  although  the  form  of  statement  appears  to 
make  it  different.  The  reason,  however,  for  the  change  is  in 
the  peculiar  nature  of  the  proposition,  together  with  the  quan- 
tification of  the  term  "kings."  If  we  observe  the  two  propo- 
sitions we  shall  discover  that  only  one  of  them  connects  the 
subject  and  the  predicate  by  means  of  the  ordinary  copula. 
But  this  is  of  minor  importance  compared  with  the  fact  that 
the  two  terms  so  connected  are,  one  of  them  an  individual  or 
singular  and  the  other  a  class  term.  This  makes  the  proposi- 
tion quantitative,  or  an  extensive  judgment,  its  intensive  mean- 
ing being  implied.  We  have  already  explained  how  all  prop- 
ositions may  be  either  intensive  or  extensive,  according  as  we 
choose  to  view  them.  This  double  conception  of  them  gener- 
ally enables  us  to  proceed  in  an  argument  without  regard  to  the 
other  fact,  which  we  remarked  of  them,  namely,  that  the 
quantity  of  intension  is  represented  in  the  reverse  order  of  that 
in  extension.  In  other  words,  according  to  Hamilton  and  oth- 
ers, judgments  of  extension  represent  the  subject  as  con- 
tained in  the  predicate,  and  judgments  of  intension  represent 
the  predicate  as  contained  in  the  subject.  Practically,  how- 
ever, this  inversion  of  the  order  of  comprehension  has  no  im- 
portance so  long  as  the  judgments  are  convertible  from  the 
intensive  to  the  extensive,  and  vice  versa,  by  the  mere  thinking 
of  those  qualities  and  without  a  change  in  the  order  of  the 
terms.  This  is  because  the  propositions  usually  chosen  for 
illustration  are  simple  ones.  But  the  proposition,  "  The  Di- 
vine Law  commands  us  to  honor  kings,"  is  not  a  simple  one 
in  the  ordinary  logical  sense.  In  the  first  place,  it  is  an  in- 
tensive judgment.  The  predicate,  "  commands  us  to  honor 
kings,"  is  not  a  class  term,  but  expresses  a  quality  inherent  in 
the  "Divine  Law."  But  nevertheless,  as  explained  before,  it 
can  be  immediately  transformed  into  an  extensive  judgment 
by  saying,    "  The  Divine  Law  is  that  which  commands,"  etc. 


286  ELEMENTS  OF  LOOIO 

This  is  the  form  in  which  the  diagrams  represent  it.  But  at 
the  same  time  it  is  apparent,  both  froni  the  reasoning  and  from 
the  diagrams,  that  the  inference  does  not  depend  upon  this 
feature  of  its  intension  and  extension.  Hie  quantitative  element 
which  determines  the  reasoning  is  in  connection  with  the  term 
"kings."  It  is  this  that  renders  the  proposition  logically 
complex.  There  is  a  double  quantification  of  concejitions. 
But  being  an  intensive  proposition,  Avith  the  assertion  or  im- 
plication that  "  kings  "  is  to  be  taken  in  its  extension,  and 
because  the  whole  reasoning  depends  upon  the  latter,  we  may 
ignore  the  relation  of  intension  and  extension  expressed  be- 
tween "  Divine  Law  "  and  "  commands,"  etc.,  because  this  is 
reproduced  in  the  conclusion,  and  reconstruct  the  proposition 
formally  so  as  to  express  the  way  in  which  the  mind  deals 
with  it.  In  other  words,  we  transform  it  from  its  intensive 
form  into  an  extensive  judgment  like  the  other  premise  in  the 
syllogism,  and  having  the  quantitative  conceptions  implicitly 
used  in  the  process  explicitly  stated  in  the  order  to  indicate 
the  conformity  of  the  reasoning  to  the  regular  laws  of  the 
syllogism.  The  proposition  "  The  Divine  Law  commands  us 
to  honor  kings,"  becomes,  considering  that  "  kings  "  is  univer- 
salized in  its  meaning,  "  All  kings  are  those  whom  the  Divine 
Law  commands  us  to  honor."  The  Mood  and  Figure  is  appar- 
ent from  this  construction,  and  it  is  brought  out  by  observing 
that  the  judgment  is  an  extensive  one,  with  its  quantity,  so  far 
as  it  affects  the  reasoning,  in  one  of  the  terms,  and  not  in  the 
whole  predicate. 

Now,  as  we  have  remarked,  a  large  amount  of  our  ordinary 
reasoning  is  of  this  kind,  namely,  is  reasoning  with  proposi- 
tions involving  complexities  of  intension  and  extension  that 
require  analysis  to  bring  them  under  the  regular  rules.  As 
indicated,  we  do  this  by  making  both  premises  extensive  prop- 
ositions. Even  if  one  of  them  is  intensive  it  must  be  capable  of 
an  extensive  conception  or  expression  which  indicates  the  proper 
relation  of  inclusion  or  exclusion  between  it  and  the  other  prem- 
ise, and  this  is  making  it  in  effect  extensive. 

That  this  is  a  common  state  of  matters  in  reasoning  can  be 


MATHEMATICAL  AND  OTHER  REASONING        2S7 

seen  by  more  examples  than  the  one  we  have  been  discussing. 
We  shall  give  some,  both  in  the  valid  and  the  invalid  form. 
Take  the  first  : 

Men  show  a  disposition  to  respect  the  brave. 

Leonidas  was  brave. 

Therefore  men  show  a  disposition  to  respect  Leonidas. 

This  instance  is  precisely  like  the  previous  one,  and  appears 
to  be  of  the  second  Figure,  but  is  resolved  into  the  first  by 
the  same  process,  and  for  the  same  reasons  as  before.  The 
major  premise  becomes  "the  brave  are  those  whom  men  show 
a  disposition  to  respect."  That  the  reduction  is  dependent 
upon  the  peculiarly  intensive  nature  of  the  proposition  is  ap- 
parent from  the  following  instance  which  has  the  same  form, 
but  does  not  resolve  so  easily  : 

Men  show  a  disposition  to  be  noble. 

Lincoln  was  noble. 

Therefore  men  show  a  disposition  to  be  Lincoln. 

This  is  manifestly  absurd.  But  when  reconstructed  it  gives 
the  conclusion,  "  Lincoln  is  one  of  those  whom  men  show  a 
disposition  to  be,"  which  is  rational  enough.  "Why  does  it 
not  appear  so  in  the  first  case,  where  we  seem  to  proceed  as 
before  ? 

The  answer  to  this  is  found  in  the  difference  of  import  be- 
tween the  use  of  be  and  a  transitive  verb.  "  To  honor  kings," 
"  to  respect  the  brave,"  are  intensive  conceptions  whose  ex- 
tension can  be  brought  out  only  as  in  the  diagrams,  by  revers- 
ing the  order  of  expression,  as,  "Kings  are  to  be  honored," 
"The  brave  are  to  be  respected."  But  "  to  be  brave,"  "  to  be 
a  king,"  may  be  intensive  or  extensive,  as  we  choose  to  regard 
them,  and  without  any  alteration  of  order.  In  the  former  the 
two  qualities  can  be  made  to  coincide  only  by  the  inversion 
we  have  indicated,  and  hence  such  intensive  conceptions  or 
propositions  must  be  transformed  in  order  to  show  the  formal 
process  of  reasoning  actually  involved  in  their  use. 


2S8  ELEMENTS  OF  LOO 10 

An  instance  of  an  invalid  form  may  be  produced  in  order  to 
confirm  in  a  negative  way  the  reduction  we  have  made  : 
All  oxen  are  animals. 

The  Levitic  law  permits  us  to  use  oxen  for  food. 
Therefore  the  Levitic  law  permits  us  to  use  animals  for  food. 

If  in  the  conclusion  we  mean  "some  animals,"  the  reason- 
ing is  valid,  but  if  we  mean  "  all  animals,"  as  we  have  ex- 
plained to  be  usual  in  such  propositions,  the  inference  is  not 
valid.  We  have  distributed  the  term  "  animals  "  in  the  con- 
clusion when  it  is  not  distributed  in  the  premises.  But  it  is 
noticeable  that  the  form  of  the  syllogism  is  that  of  AAA  of  the 
first  Figure,  and  yet  is  not  valid  except  on  the  condition  that 
we  say  or  mean  "  some  animals"  in  the  conclusion.  When  re- 
duced it  becomes  in  reality  AAA  of  the  third  Figure,  which  is 
a  case  of  illicit  minor  term.     Thus, 

All  oxen  are  beings  which  the  Levitic  law  permits  us  to  use 
for  food. 

All  oxen  are  animals. 

Therefore  all  animals  are  beings  which  the  Levitic  law  per- 
mits us  to  use  for  food. 

Here  the  illicit  minor  is  quite  apparent.  But  if  we  had  said 
"  Some  animals  are  beings,"  etc.,  the  conclusion  would  be 
valid,  and  it  would  explain  why  to  have  said  or  meant  "  some 
animals  "  in  the  first  instance  would  have  been  valid  reason- 
ing. As  is  usual  in  the  third  Figure,  it  does  not  matter 
which  of  the  propositions  is  taken  for  the  major  premise. 
The  conclusion  will  be  substantially  the  same  in  all  cases. 
We  placed  them  in  the  above  order  solely  for  the  convenience 
of  getting  the  conclusion  in  an  order  that  would  disj^ense 
with  conversion  for  comparing  it  with  the  original. 

The  first  of  the  cases  can  be  solved  also  according  to  the 
same  principles.  The  proposition,  "  The  Persians  worship  the 
sun,"  can  be  reduced  to  "  The  sun  is  the  object  which  the  Per- 
sians worship."  This  will  make  the  syllogism  AAA  of  the 
third  Figure  instead  of  the  first,  and  the  conclusion  will  be, 
"  The  object  which  the  Persians  worship  is  a  thing  insensible." 


MATHEMATICAL  AND   OTHER  REASONING        2S9 

But  there  is  no  necessity  for  this  reduction  in  this  instance, 
because  the  middle  term  is  singular  and  the  predicate  of  the 
major  premise  is  made  indefinite  or  left  unquantified  by  the 
particle  "a,"  which  prevents  it  from  being  universalized  in 
the  conclusion,  and  which  has  the  effect  of  equating  it  with 
the  subject  in  the  minor  premise.  Were  it  universalized  in  the 
conclusion  we  should  have  an  instance  of  the  same  fallacy  as 
that  which  we  have  just  exposed.  But  the  case  is  an  inter- 
esting one  as  showing  the  use  of  substitution  whenever  singu- 
lar terms  are  employed,  and  the  peculiar  influence  which  may 
be  exercised  by  the  particles  "a"  or  "an,"  and  "  the,"  the 
former  denoting  an  individual  which  is  at  the  same  time  a 
part  of  a  class  and  the  substitutive  equivalent  of  the  term 
which  it  defines.  The  latter  often  denotes  that  the  term 
which  it  agrees  with  is  quantified  universally,  or  that  a  certain 
definite  number  of  objects  is  considered,  which  enables  us  to  re- 
sort to  substitution.  They  do  not,  however,  alter  the  form  of 
a  proposition,  and  do  not  always  produce  the  effect  described. 
The  meaning,  as  affected  by  them,  is  a  material  factor  of  the 
proposition.  But  it  is  important  to  consider  it  for  the  reason 
that  it  may  make  the  reasoning  actually  valid  in  cases  where, 
tested  by  purely  formal  laws  it  would  appear  invalid. 

In  all  these  instances  of  complex  syllogisms  we  have  clear 
illustrations  of  an  apparent  resemblance  to  mathematical  rea- 
soning in  that  there  is  at  least  an  apparent  disregard  of  the 
Figure  of  the  syllogism.  But  with  the  reality  of  this  disre- 
gard denied  the  illusion  is  dispelled,  and  we  have  illustrations 
of  reasoning  in  which  traduction  is  not  possible  unless  the 
terms  used  are  singular.  The  contrast,  therefore,  between 
purely  quantitative  reasoning,  in  which  the  subject  and  pred- 
icate are  identical  in  extension,  and  qualitative  reasoning 
combined  with  the  quantitative,  where  there  is  a  disparity  of 
extension  between  subject  and  predicate  could  not  be  better 
brought  out  than  in  these  cases.  And  as  much  of  our  rea- 
soning is,  perhaps,  of  the  type  we  have  just  been  considering,  we 
see  the  liability  to  fallacy  incident  to  it  because  of  its  variation 

from  the  form  where  certitude  and  assurance  are  guaranteed. 
19 


CHAPTER  XXI. 

THE   LAWS   OF   THOUGHT 

The  Laws  of  Thought  do  not  require  any  elaborate  treat- 
ment in  an  elementary  treatise  upon  Logic,  but  the  manner  in 
which  they  have  beeu  assumed,  or  in  which  they  underlie  all 
our  reasonings  makes  it  necessary  to  state  them  and  their 
meaning  very  briefly.  We  have  already  explained  what  a 
"  law  of  thought "  means,  in  our  statement  that  it  denotes  the 
uniform  way  in  which  we  think  and  must  think.  In  all  our 
reasoning  we  take  these  laws  for  granted.  They  are  condi" 
tions  of  our  reasoning  and  of  the  relation  expressed  between 
subject  and  predicate,  antecedent  and  consequent,  in  proposi- 
tions. We  do  not  require  to  announce  them  as  premises  in 
our  processes  of  transition  from  proposition  to  proposition, 
because  they  are  either  universally  assumed  without  question, 
or  they  are  the  conditions  of  the  formal  and  material  truth  of 
the  data  themselves,  which  it  is  not  the  business  of  formal 
Logic  to  investigate.  Besides,  they  are  of  that  axiomatic 
nature  which  renders  it  necessary  to  admit  them  before  we 
could  construct  an  objection  or  an  argument  against  them. 
We  do  not  require,  therefore,  to  investigate  them  to  determine 
their  validity,  but  only  to  state  what  they  are,  their  meaning 
and  their  functions. 

The  Laws  of  Thought  may  be  divided  into  two  classes,  the 
Primary  or  Fundamental,  and  the  Secondary  or  Derived.  The 
primary  laws  are  those  which  regulate  all  thought,  whatever, 
whether  of  Conception,  Judgment,  or  Reasoning.  The  second- 
ary are  simply  those  modified  forms  of  the  primary  laws  which 
are  formulated  in  a  particular  way  to  suit  the  contingencies  of 
syllogistic  reasoning.     We  shall  consider  them  in  their  order. 

1st,  The  Primary  Laws. — As  defined,  they  are  the  funda- 


THE  LAWS  OF   THOUGHT  291 

mental  laws  of  all  thinking  ;  that  is,  of  conceiving  percepts, 
concepts,  and  judgments  in  relation  to  each  other.  There 
are  four  of  these  general  laws,  Ttie  Law  of  Identity,  The  Law 
of  Contradiction,  TJie  Law  of  Excluded  Middle,  and  Tlie  Law  of 
Sufficient  Reason. 

1.  The  Law  of  Identity. — This  law  exj>resses  the  right  of 
the  mind  to  affirm  that  a  thing  is  identical  with  itself.  Thus, 
A  is  A,  or  "  Whatever  is,  is,"  that  is,  any  existence  is  equal  to 
itself.  This  is  the  usual  form  of  statement  for  the  law.  But 
the  law  is  the  general  principle  at  the  foundation  of  all  affirm- 
ative judgments,  whether  the  subject  and  predicate  are  quanti- 
tatively or  qualitatively  equal  or  not.  Hence  there  are  two 
kinds  of  identity,  absolute  or  total,  and  relative  or  partial  iden- 
tity. Absolute  identity  is  represented  by  the  truistic  or  tau- 
tological proposition,  where  both  the  form  and  the  matter  of 
the  terms  in  subject  and  predicate  are  the  same.  Eelative  or 
partial  identity  is  that  of  the  ordinary  proposition  where  there 
is  a  difference  of  extension  between  subject  and  predicate,  or 
where  the  form  of  the  terms  makes  it  possible  to  identify  them 
with  others  also.  Thus  to  illustrate  both  instances  :  "Man  is 
man,"  "Animals  are  animals,"  are  cases  of  absolute  identity 
in  which  a  conception  can  be  affirmed  to  be  equal  to  itself. 
But  of  partial  identity  we  have  "  Man  is  an  animal,"  "  Horses 
are  quadrupeds."  In  these  instances  there  are  certain  ele- 
ments of  identity,  but  the  terms  are  not  convertible  with  each 
other  ;  that  is,  the  propositions  cannot  be  converted  simply. 

From  what  lias  just  been  said  it  might  be  inferred  that  ex- 
clusive propositions  and  definitions,  as  well  as  synonymous 
terms,  are  illustrations  of  absolute  identity.  This  is  true  ;  but 
it  is  identity  of  matter  and  not  of  form.  The  terms  are  such  as 
can  be  used  in  other  than  truistic  propositions. 

2.  Tue  Law  of  Contradiction. — This  law  Hamilton  observes 
should  be  called  the  law  of  non-contradiction,  because  it  de- 
notes that  an  object  cannot  be  affirmed  to  be  what  it  is  not. 
It  is  sometimes  defined  as  denoting  that  a  thing  cannot  exist 
and  not  exist  at  the  same  time  ;  or  that  it  cannot  be  affirmed 
to  be  one  thing  and  its  opposite  or  contradictory  at  the  same 


292  ELEMENTS  OF  LOGIC 

time.  This  law  is  the  complement  of  the  law  of  identity,  and 
may  be  said  to  be  the  same  law  for  thought  as  the  law  of  im- 
penetrability is  for  matter.  It  is  the  principle  which  deter- 
mines the  relations  and  inferences  in  the  Square  of  Opposition 
and  the  drawing  of  negative  conclusions  in  the  syllogism. 

3.  The  Law  of  Excluded  Middle. — This  law  denotes  that 
of  two  contradictions  only  one  can  be  true.  Thus  of  the 
propositions,  "  Man  is  an  animal,"  and  "  Man  is  not  an  ani- 
mal," only  one  can  be  true.  They  are  supposed  to  represent 
a  completely  dichotomous  division  of  all  objects  into  two 
contradictory  classes,  and  to  achieve  this  the  subject  and 
predicates  must  be  so  related  that  if  the  law  of  identity  ap- 
plies to  the  relation  in  one  case,  that  of  contradiction  ap- 
plies in  the  other.  Hence  only  one  can  be  true,  and  the  other 
must  be  false.  The  law  is  a  combination  of  the  first  two 
laws,  those  of  identity  and  contradiction,  and  is  at  the  basis  of 
disjunctive  propositions  and  syllogisms. 

4.  The  Law  of  Sufficient  Keason. — This  law  is  briefly  de- 
fined as  denoting  that  every  phenomenon,  event,  or  relation 
must  have  a  sufficient  reason  or  cause  for  being  what  it  is. 
There  is  some  dispute  about  the  right  to  regard  this  law  as  a 
law  of  reasoning,  and  it  is  certain  that  it  seems  more  appro- 
priate an  assumption  or  postulate  for  the  physical  sciences 
than  for  the  logical.  But  it  nevertheless  dominates  certain 
modes  of  thought  which  are  occupied,  not  with  the  identity  or 
non-identity  of  objects  or  concepts,  but  with  their  connection. 
"We  shall  not  discuss  the  merits  of  this  question,  since  it  does 
not  belong  to  the  elementary  plan  of  the  present  work  to  do 
so.  Hence,  we  shall  only  explain  the  meaning  attached  to  the 
law  by  those  who  regard  it  as  a  law  of  thought. 

The  law  has  been  formulated  to  mean  that  we  should  affirm 
or  infer  nothing  without  a  reason  or  ground.  This  reason  or 
ground  is  the  condition  upon  which  the  truth  of  a  proposition 
or  the  reality  of  an  event  depends.  The  condition  may  be 
called  the  antecedent  and  the  resultant  the  consequent. 
Hence  the  law  will  appear  to  determine  the  relation  expressed 
in  hypothetical  propositions  and  syllogisms.     But  as  these  are 


THE  LAWS  OF  THOUGHT  293 

reducible  to  the  categorical  form  the  law  must  either  be  re- 
duced to  that  of  identity,  or  be  found  to  apply  to  categorical 
judgments.  Perhaps  it  is  the  dependence  of  all  individual 
truths  upon  general  truths  or  principles  that  represents  the 
application  of  this  law  as  well  as  that  of  identity  in  all  ordi- 
nary propositions  and  reasoning.  If  so,  it  can  have  an  inde- 
pendent place  in  Logic.  But  not  intending  to  decide  this 
question,  pro  or  con,  we  must  be  content  with  recognizing 
what  is  frequently  regarded  as  a  law  of  thought,  and  what 
certainly  expresses  a  relation  of  dependence  quite  fundamen- 
tal to  our  mental  processes. 

2d.  The  Secondary  Laws. — These  are  simply  derivatives 
of  the  primary  laws  of  Identity  and  Contradiction.  They  are 
often  called  the  axioms  of  Logic,  and  are  simply  formulas  for 
justifying  the  inferences  in  formal  reasoning.  They  are  four, 
two  of  them  for  immediate  inference  and  two  of  them  for  me- 
diate reasoning.  After  what  has  been  said  in  previous  chap- 
ters they  may  be  stated  without  further  explanation. 

1.  If  two  concepts  agree  in  one  relation,  they  may  be 
stated  to  agree  in  the  converse  relation. 

2.  If  two  concepts  differ  or  contradict  in  one  relation,  they 
will  disagree  or  contradict  in  the  converse. 

These  rules  regulate  the  process  of  immediate  inference  or 
Conversion,  and  their  principle  is  simply  assumed  in  every 
case  of  convertible  and  non-convertible  terms. 

3.  If  two  terms  agree  with  one  and  the  same  third  term, 
they  agree  with  each  other. 

4.  If  of  two  terms  one  agrees  and  the  other  disagrees  with 
one  and  the  same  third  term,  they  do  not  agree  with  each 
other. 

It  must  be  observed,  however,  that  the  latter  two  axioms 
are  applicable  in  their  purity  and  in  an  unqualified  sense, 
only  to  mathematical  syllogisms,  or  reasoning  with  universally 
or  definitely  quantified  subjects  and  predicates.  Hence  for 
ordinary  syllogisms  they  have  to  be  modified  by  the  following 
rule  or  law  : 

5.  The  terms   which    agree  or  disagree  in  the  conclusion 


294  ELEMENTS  OF  LOGIC 

must  have  no  greater  or  the  same  distribution  as  in  the  prem- 
ises, and  they  can  agree  or  disagree  only  when  the  middle 
term  is  properly  distributed  in  the  premises. 

This  last  law  is  to  provide  against  the  commission  of  the 
fallacies  of  Illicit  Middle,  Illicit  Minor,  and  Illicit  Major  terms, 
when  the  third  and  fourth  laws  have  been  conformed  to  in 
their  qualitative  relations.  The  fifth  lawr  or  rule  specifies  the 
quantitative  conditions  affecting  the  conclusion. 


CHAPTER  XXTL 

INDUCTIVE    REASONING 

1st.  The  Nature  of  Induction. — It  has  been  usual  to  de- 
fine Induction  in  a  manner  contrasted  with  Deduction.  But 
there  are  some  peculiarities  in  connection  with  the  meaning 
of  the  term  which  must  be  considered  before  we  define  the 
process  with  which  we  are  at  present  concerned.  Usage  has 
not  given  the  term  a  uniform  conception,  such  as  belongs 
cprite  generally  to  the  word  "Deduction."  All  classes  of 
thinkers  are  tolerably  agreed  in  regard  to  the  process  of  de- 
duction, whether  they  are  logicians  or  scientific  investigators. 
It  is  assumed  to  be  the  process  of  finding  the  proof  of  a  par- 
ticular truth  in  a  general  principle  or  proposition  already  con- 
taining it,  explicitly  or  implicitly.  Induction  is  often  de- 
scribed as  the  reverse  of  this  process,  namely,  as  inferring 
general  truths  from  the  particular.  But  this  is  not  the  only 
conception  of  the  term,  and  it  is  because  this  conception  is 
not  the  only  one  in  use  that  there  is  so  little  agreement  about 
the  nature,  functions,  and  importance  of  induction.  It  will, 
therefore,  be  necessary  to  examine  the  several  imports  of  the 
term  in  order  to  make  possible  a  true  conception  or  theory  of 
the  process  denoted  by  it. 

There  are  three  different  ajyplications  of  the  term  "Induc- 
tion," which  are  generally  assumed  to  mean  the  same  thing. 
To  explain  what  they  are  we  have  to  produce  the  usual  di- 
visions of  the  subject,  wdiich  are  the  so-called  kinds  of  induc- 
tion. They  are  "  Perfect  Induction  "  and  "  Imperfect  Induc- 
tion." The  first  meaning  of  the  term  applies  to  the  first  kind, 
and  the  other  two  are  modifications  of  what  is  implied  in  im- 
perfect induction.  We  would  not  suspect  a  difference  of 
meaning  from  this  general  fact  alone,  but  if  we  examine  care- 


296  ELEMENTS  OF  LOGIC 

fully  the  illustrations  chosen  to  describe  the  nature  of  the 
process  as  thus  distinguished  into  different  kinds,  we  shall 
discern  very  clearly  the  great  differences  of  real  meaning  at- 
taching to  the  term.  Thus  "Perfect  Induction  "  is  simply  an 
enumeration  of  the  particulars  w)ti<-li  form  a  class.  It  is  the 
process  which  characterized  the  method  of  Socrates  in  reach- 
ing his  definitions,  and  which  Aristotle  remarked  was  a  new 
method  compared  with  the  argumentation  of  his  predecessors. 
An  example  of  perfect  induction  is  the  following :  "  Mercury 
revolves  on  its  axis  ;  so  do  Venus,  the  Earth,  Mars,  Jupiter,  Sat- 
urn, and  Neptune.  But  these  are  all  the  jnanets,  and  therefore 
all  the  planets  revolve  on  their  axes."  Although  this  is  stated 
in  the  form  of  reasoning,  it  is  not  reasoning  at  all.  This  fact 
is  apparent  in  the  nature  of  the  conclusion,  which  is  that  "  All 
the  planets  revolve  *on  their  axes,"  not  on  the  axis  of  Mercury, 
although  the  same  thing  would  be  true  if  we'  had  said  "  on  the 
axis  of  Mercury."  But  the  special  proof  of  its  not  being  a 
case  of  reasoning  is  in  the  fact  that  the  so-called  conclusion  is 
merely  a  universal  statement  of  what  had  been  enumerated 
in  detail  in  the  premises.  We  are  supposed  to  have  observed 
the  individual  fact  that  "  Mercury  revolves  on  its  axis,"  and 
then  again  that  "Venus  revolves  on  its  axis,"  and  so  on 
throughout  the  entire  number  of  planets.  Hence  when  we 
say,  "  All  the  planets  revolve  on  their  axes,"  we  but  univer- 
salize our  particular  observations — we  use  the  terms  "  all 
planets"  as  an  economical  device  to  avoid  repeating  the 
proper  name  of  each  planet.  But  we  do  not  infer  anything, 
or  reason  from  one  proposition  to  another.  We  do  not  es- 
tablish any  new  connections  of  thought  by  the  process,  as  we 
do  in  syllogistic  reasoning,  as  already  explained,  but  we  only 
generalize  what  we  had  observed  in  detail.  It  is  precisely  the 
same  with  all  enumerations  of  individuals  or  particulars  into 
a  whole  or  class  with  a  general  name  denoting  those  enumera- 
tions only.  They  may  be  called  "  Inductions  "  if  we  choose 
so  to  name  them  ;  but  they  are  not  reasoning.  They  are  only 
generalizations  as  opposed  to  or  distinct  from  reasoning,  while 
the  term  "  Induction,"  as  now  used  by  logicians,   denotes  a 


INDUCTIVE  REASONING  297 

process  of  inference  or  reasoning  of  some  kind.  It  is  quite 
generally  agreed  since  the  time  of  Bacon  that  the  so-called 
"Perfect  Induction"  is  not  properly  called  "Induction" 
because  it  is  not  a  mode  of  reasoning.  It  has  been  the 
name  for  the  Socratic  process  of  obtaining  universal  concep- 
tions and  definitions.  But  the  contingencies  of  the  growth 
of  knowledge  and  the  demand  for  a  method  which  would  take 
the  place  of  the  Aristotelian  Logic  suggested  the  term  "  In- 
ductive" as  opposed  to  "Deductive,"  and  the  rejection  of 
"  Perfect  Induction  "  on  the  ground  that  it  was  not  ratiocina- 
tive  in  its  nature,  implied  that  Induction  must  be  a  process  of 
reasoning  in  order  to  compare  it  with  Deduction. 

This  second  general  meaning  is  the  more  important  of  the 
two,  and  was  called  "Imperfect  Induction"  because  the  con- 
clusion contained  more  than  the  premises.  Thus  if  I  had  in- 
ferred that  "  All  the  planets  revolve  about  their  axes,"  from 
the  mere  fact  that  one  of  them  did  so,  I  should  have  drawn 
an  inductive  inference.  I  should  not  in  this  case  have  merely 
generalized  the  particulars  of  my  observation  or  experience, 
but  have  conjectured  or  inferred  that  what  was  true  of  one 
case  would  turn  out  to  be  true  of  all  the  objects  known  upon 
other  grounds  to  belong  to  the  same  class.  But  this  conclu- 
sion has  no  definite  certainty  such  as  the  mind  desires,  and 
hence  to  give  this  conjecture  greater  probability  I  must  vary 
my  observation  of  facts  in  connection  with  the  several  planets, 
and  find  whether  they  agree  or  disagree  with  my  supposition. 
If,  for  instance,  I  observed  that  certain  of  them  presented  an 
absolutely  invariable  appearance,  such  as  a  particular  spot  al- 
ways in  sight  and  in  the  same  place,  the  fact  would  be  at  least 
a  presumption  against  the  supposition  of  the  planet's  axial 
revolution.  On  the  other  hand,  if  the  spot  presented  certain 
regular  changes  of  position  and  periodical  disappearance  and 
reappearance,  the  fact  would  be  in  favor  of  the  hypothesis. 
This  mode  of  repeating  and  varying  observations  or  experi- 
ments in  the  case  of  the  experimental  sciences,  according  to 
certain  methods,  which  are  called  the  "  Method  of  Agree- 
ment," the  "  Method  of  Difference,"  the  "  Method  of  Concom- 


298  ELEMENTS  OF  LOGIC 

itant  Variations,"  etc.,  has  been  called  the  Inductive  Method  in 
general,  as  a  mode  of  ascertaining  certain  truths  in  a  manner 
quite  distinct  from  the  ordinary  syllogistic  and  deductive 
reasoning.  This  is  the  third  meaning  of  the  term,  with  which 
a  theory  of  Induction  has  to  reckon.  It  remains  for  us  to  se- 
lect which  of  them  we  are  to  deal  with  under  the  heading  of 
this  chapter.  It  is  the  confusion  of  all  three  under  the  same 
term  that  leads  to  the  uncertainties  about  the  process  itself. 

It  will  conduce  to  clearness  in  the  discussion  of  Induction 
if  we  sketch  briefly  the  history  of  the  general  meaning  of  the 
term,  and  indicate  specially  the  implication  carried  along  with 
it  which  is  no  necessary  part  of  its  import  as  describing  a  logi- 
cal process.  This  latter  fact  is  not  sufficiently  taken  into 
account  b}r  logicians  when  treating  of  the  subject,  and  yet  it 
involves  grave  consequences  to  their  theory  of  Induction. 

The  most  general  meaning  of  the  term  "  Induction,"  as 
used,  especially  by  popular  writers,  since  the  time  of  Bacon,  is 
that  of  a  process  which  adds  to  our  knowledge.  The  accusation 
which  Bacon  and  his  admirers  have  brought  against  the  "  de- 
ductive method,"  or  Aristotelian  Logic,  was  that  it  could  not 
give  us  our  premises,  and  therefore  neither  assured  us  of  our 
data  for  reasoning,  nor,  when  we  were  assured  of  them,  could 
it  add  anything  to  our  knowledge.  In  other  words,  as  we 
have  seen,  our  conclusion  depends  wholly  upon  knowing  our 
premises,  and  if  we  already  know  the  premises,  the  conclusion 
adds  nothing  to  what  we  know.  The  deductive  method  is, 
therefore,  useless  for  giving  us  knowledge,  say  its  opponents. 
It  only  manipulates  in  various  ways  that  which  we  are  sup- 
posed to  have  in  an  implicit,  if  not  an  explicit,  form.  How, 
then,  do  we  get  the  knowledge  wre  have  ?  How  do  we  ever 
make  any  additions  to  our  general  knowledge,  or  to  the  prem- 
ises with  which  deduction  deals  and  which  it  assumes  ? 

This  was  the  question  which  Bacon  attempted  to  answer, 
and  he  employed  the  term  "  Induction  "  to  define  the  method 
of  acquring  new  data  and  principles  of  truth.  The  Aristote- 
lian method  was  rejected  as  useless,  because  it  could  never  ad- 
vance beyond  what  was  already  given,  and  hence,  in  adopting 


INDUCTIVE  REASONING  299 

the  new  method,  which  was  to  contrast  with  the  old,  the  term 
"Induction"  took  on  the  meaning  which  that  contrast  im- 
plied, namely,  that  process  which  produced  an  increment  to  the 
knowledge  wh  ich  we  already  have  at  any  given  time.  If  we  take 
the  first  meaning-  of  the  term,  that  describing  the  Socratic  in- 
duction, and  called  "Perfect  Induction,"  we  shall  see  that  it 
involves  to  some  extent  this  implication  ;  not  that  the  gener- 
alization at  the  end  of  our  observations  expresses  any  addi- 
tion to  oar  previous  knowledge,  but  that  the  enumeration 
of  individual  experiences  or  observations,  or  perhaps  better, 
the  accumulation  of  them,  is  such  a  process.  It  is  quite 
as  apparent  that  the  inference  from  one  or  more  known 
facts  to  a  greater  number  is  an  addition,  or  the  necessary 
step  to  such  an  addition.  But  if  it  be  called  "induction" 
it  must  be  regarded  as  quite  of  a  different  nature  from  the 
process  which  we  have  just  mentioned.  It  nevertheless  re- 
sembles it  in  the  one  quality  of  providing  an  increment  to  ex- 
isting knowledge.  And  again,  the  "  methods  of  Induction  " 
which  seek  by  repeated  observatious  and  experiments  to  verify 
a  supposition  or  conjecture  already  made,  represent  means  of 
adding  to  previous  knowledge.  But  they  represent  also  some- 
thing more  than  an  inductive  inference.  They  are  compli- 
cated with  direct  experience  and  observation,  and  with  deduc- 
tive principles,  assumptions,  and  inferences,  so  that  they  are 
not  simply  cases  of  inductive  reasoning,  although  they  are  the 
proper  means  of  adding  to  our  knowledge  and  widening  our 
generalizations. 

But  Logic  does  not  immediately  treat  of  increasing  our 
knowledge,  or  of  the  material  means  for  applying  scientific 
methods.  It  has  to  do  with  thought  or  reasoning  and  with 
the  maimer  in  which  one  idea  is  inferred  from  another,  not 
with  the  complicated  methods  of  verifying  this  inference  after 
it  is  made.  Hence  in  so  far  as  "  Induction  "  is  a  logical  pro- 
cess in  contrast  with  Deduction,  we  must  confine  the  term  to  a 
certain  kind  of  reasoning,  and  not  extend  it  to  all  modes  of 
adding  to  our  previous  knowledge.  We,  therefore,  choose  the 
second  of  the  three  meanings  explained,  as  the  proper  one  to 


300  ELEMENTS  OF  Loan1 

represent  "Induction"  in  so  far  as  formal  Logic  has  to  deal 
with  it,  and  which  is  the  only  meaning  that  will  enable  us  to 
contrast  the  process  with  Deduction  or  consider  it  exclusively 
as  a  process  of  reasoning.  This  is  very  clear  in  the  case  of 
the  so-called  "Perfect  Induction,"  which  we  have  found  to  be 
simply  observation  and  generalization,  and  not  properly  rea- 
soning at  all,  and  which  is  as  much  a  condition  of  Deduction 
as  of  Induction. 

The  third  form  is  a  process  of  verification  of  conjectures,  or 
probable  inferences  already  made,  and  although  such  addi- 
tional inferences  may  be  connected  with  these  various  methods 
of  verification,  the  methods  themselves  are  not  pure  inductive 
inferences,  as  is  admitted  by  more  than  one  writer  on  Logic. 
But  to  these  points  we  shall  return  again,  when  they  may  be 
treated  more  fully.  It  has  been  iu^ortant  here  only  to  fix 
upon  the  meaning  which  the  term  "  Induction  "  is  to  have 
when  comparing  it  with  deductive  reasoning,  and  to  establish 
the  fact  that  the  notion  or  implication  so  common  with  popu- 
lar writers,  and  tacitly  admitted  by  logicians  themselves, 
namely,  that  "  Induction  "  is  any  process  of  adding  to  what 
we  already  know  at  airy  given  time,  is  not  the  main  conception 
with  which  we  have  to  do  when  considering  it  as  a  form  of 
reasoning.  It  does  imply  addition  ;  but  it  is  by  way  of  infer- 
ence, not  by  observation  or  verification. 

Having  fixed  upon  the  second  of  the  three  meanings  as  the 
proper  one  for  the  term,  so  far  as  formal  Logic  is  concerned 
with  it,  we  may  indicate,  before  defining  the  process  more 
carefully,  the  three  methods  and  their  characteristics  by  a  ter- 
minology to  some  extent  new.  The  first  we  shall  call  Observa- 
tion and  Generalization,  the  second,  Induction,  and  the  third, 
Verification  or  Scientific  Method.  We  use  the  term  "Induc- 
tion," therefore,  to  describe  a  process  of  inferring  one  truth 
from  another  in  a  manner  somewhat  different  from  Deduction, 
and  are  now  prepared  to  define  it  more  accurately. 

We  have  already  defined  Induction  as  reasoning  front  the 
particular  to  the  universal  in  contrast  with  Deduction  as  rea- 
soning from  the  universal,  to  the  particular.     It  is  also  fre- 


INDUCTIVE  REASONING  301 

quently  defined  as  reasoning  from  effects  to  causes,  from  the 
known  to  the  unknown,  from  the  actual  to  the  possible  and 
probable.  From  two  of  these  conceptions  it  is  often  supposed 
that  the  process  is  the  inverse  of  Deduction,  and  this  is  true 
in  some  respects,  but  not  in  all.  For  instance,  to  argue  from 
the  known  to  the  unknown,  or  from  the  actual  to  the  possi- 
ble and  probable,  is  not  the  inverse  of  Deduction,  for  if 
it  were,  we  should  be  obliged  to  regard  the  latter  as  reasoning 
from  the  unknown  to  the  known,  or  from  the  possible  to  the 
actual,  a  process  which  is  the  very  opposite  of  the  real  case. 
Induction  is  only  the  inverse  of  Deduction  in  certain  respects, 
and  these  are  with  reference  to  the  extension  of  the  concep- 
tions involved.  In  one  we  narrow  our  conceptions  as  we  pro- 
ceed to  the  conclusion,  and  in  the  other  we  widen  them.  In 
other  respects  we  cannot  say  that  the  two  processes  are  the 
inverse  of  each  other,  but  only  that  they  are  different  from 
each  other.  Thus  in  Deduction  we  argue  from  the  known  to 
either  the  known  or  the  iinknown  ;  to  the  known,  if  the  con- 
clusion is  an  actually  known  fact,  but  not  seen  in  all  its  rela- 
tions to  general  principles  until  these  relations  are  enun- 
ciated ;  and  to  the  unknown,  if  the  conclusion  happens  to  be 
a  truth  implicitly  contained  in  the  premises,  but  not  explicitly 
realized  in  the  consciousness  of  the  reasoner  or  the  hearer 
until  stated.  But  it  is  never  a  process  of  reasoning  from  the 
unknown  to  the  known.  Both  Induction  and  Deduction, 
therefore,  appear  to  be  reasoning  from  the  known  to  some- 
thing else.  But  there  are  two  differences  between  them.  In 
the  first  place,  the  known  data  are  different  as  syllogistic  mat- 
ter. In  Deduction  the  known  is  either  a  universal  principle 
in  the  abstract,  or  certain  universally  known  concrete  facts 
which  will  enable  us  to  affirm  the  same  thing  of  all  individ- 
ual instances  included  under  the  universal.  In  the  second 
place,  it  is  not  strictly  true  in  either  of  them  that  we  argue  to 
the  unknown.  In  Deduction  we  argue  to  what  is  necessarily 
included  in  the  nature  of  the  premise,  whether  known  or  un- 
known, explicit  or  implicit.  In  Induction  we  argue  to  the 
possible  and  probable  from  known  facts,  not  principles,  and 


302  UI  KM  UN 'IS   OF   LOGIC 

so  to  more  universal  truths,  which  may  in  their  turn  become 
deductive  data  when  verified.  These  facte  explain  why  it  is 
best  to  define  Induction  as  proceeding  from  the  particular  to 
the  general  or  universal,  or  from  effect  to  cause.  The  process 
gets  the  last  conception  of  its  nature  from  the  frequency  with 
which  it  is  occupied  in  determining  the  causes  of  known 
phenomena,  which  can  never  be  ascertained  deductively  be- 
cause more  than  one  cause  may  produce  the  same  known  ef- 
fect, and  when  the  cause  is  known  we  can  directly  know  what 
the  effect  will  be.  But  as  our  knowledge  begins  with  matters 
of  fact  we  usually  have  to  argue  to  their  probable  causes  by 
inductive  inference  or  conjecture,  no  doubt  after  some  ac- 
cumulated experience  and  observation,  and  then  verify  our 
inference  by  additional  methods. 

It  is  sometimes  said  that  Induction  is  reasoning  from  par- 
ticular to  particular,  and  some  writers  on  Logic  admit  or  as- 
sert that  nearly  all  the  reasoning  of  common  life  is  of  this 
order.  It  is  certainly  true  that  we  do  reason  from  one  partic- 
ular instance,  or  set  of  instances,  but  when  we  examine  into  the 
case  more  closely,  the  result  or  conclusion  in  its  real  meaning 
is  a  universal  broader  than  our  original  premise  and  containing 
both  what  we  had  reasoned  from  as  a  particular  and  what  we 
had  reasoned  to  as  a  particular.  This,  I  think,  will  be  clear, 
when  illustrated,  to  all  who  study  the  process.  But  in  order 
to  include  such  cases  in  the  definition,  Inductive  reasoning  may 
be  denned  as  reasoning  from  ivhat  is  known  in  a  certain  fact, 
or  facts,  to  the  possible  or  probable  truth  of  the  same  thing  in 
of  Iter  facts  where  it  has  not  been  observed  or  proved,  or  is  rea- 
soning from  actual  to  necessary  connection. 

As  an  illustration  of  this  inductive  reasoning  we  may  take 
the  case  of  discovering  gravitation  by  Sir  Isaac  Xewton. 
The  story  about  his  having  been  moved  to  the  discovery  by  a 
falling  apple  is  probably  legendary,  but  it  may  be  used  as  if  it 
were  a  fact  because  it  is  to  the  purpose  and  might  have  been 
employed  by  him  as  an  illustration.  But  supposing  his  atten- 
tion to  have  been  arrested  by  the  fall  of  an  apple  while  reflect- 
ing on  the  position  of  the  heavenly  bodies,  he  had  before  him 


INDUCTIVE  REASONING  303 

several  facts.  There  is,  first,  the  fixed  and  suspended  position 
of  the  apple  before  it  breaks  loose  from  the  tree.  Then  there 
is  its  falling  to  the  earth  under  the  influence  of  its  weight. 
The  supposition  that  it  is  attracted  by  the  earth  is  another  fact 
accounting  for  the  weight  of  the  apple.  A  fourth  fact  is  that 
the  moon,  the  sim,  the  planets,  and  other  heavenly  bodies  are 
suspended  in  space,  somewhat  in  the  same  relation  to  the  earth 
as  the  apple  which  we  are  considering.  But  it  is  not  known 
that  any  force  from  the  earth  is  exerted  to  hold  them  in  their 
position,  or  create  in  them  a  tendency  to  fall  toward  it.  But 
assuming  their  relation  to  the  earth  and  the  attraction  which 
caused  the  apple  to  fall,  we  may  infer  the  possibility  that  the 
same  force  of  attraction  extends  to  the  moon,  sun,  etc.,  inas- 
much as  we  can  see  no  reason  in  the  nature  of  space  to  pre- 
vent this  action-  As  in  the  case  of  the  moon,  for  example,  the 
resemblance  between  it  and  the  apple,  in  regard  to  relative  po- 
sitions, was  such  that  it  was  natural  to  infer  from  the  attrac- 
tion exerted  upon  the  apple  that  the  moon  was  similarly 
affected,  and  so  the  other  planets,  by  reason  of  their  likeness 
to  the  moon  in  qualities  concerning  the  matter  at  issue.  But 
there  was  one  circumstance  in  the  case  that  prevented  the  in- 
ference from  being  verified  by  observation.  It  was  the  fact 
that  the  moon  and  the  planets,  with  their  satellites,  did  not  fall 
toward  the  earth,  or  that  each  did  not  fall  to  the  body  toward 
which  it  gravitated.  In  the  case  of  the  apple  there  was  no  diffi- 
culty, because  as  soon  as  it  was  released  from  its  support  on 
the  tree,  the  phenomenon  of  its  fall  was  an  observable  fact,  and 
attraction  was  presumed  to  be  the  cause.  But  the  moon  did 
not  perceptibly  move  toward  the  earth,  although  it  was  sus- 
pended in  space,  and  without  visible  support.  Observa- 
tion, therefore,  could  do  nothing  directly  in  producing  or  ver- 
ifying the  belief  that  attraction  was  exerted  upon  the  moon, 
but  the  belief  was  an  inference  from  resemblances  of  relation 
and  material  qualities  to  a  resemblance  in  one  other  quality 
already  known  of  the  apple  and  the  earth.  Thus  far  the  in- 
ference, however  expresses  no  other  degree  of  certitude  than 
a  possibility,  and  it  remains  to  ascertain  how  a  greater  degree 


304  ELEMENTS  OF  LOGIC 

might  be  given  it  in  connection  with  the  same  or  the  same 
kind  of  reasoning. 

There  was  another  known  circumstance  in  connection  with 
the  case  which  may  have  helped  to  suggest  the  inference,  and 
which  certainly  gives  it  greater  probability.  The  motion  of 
the  planets  about  the  sun,  and  of  the  satellites  about  their  re- 
spective planets  in  elliptical  orbits,  was  an  admitted  fact  at 
the  time  of  Newton.  Now  he  knew  that  such  motion  involved 
the  existence  of  a  tendency  on  the  part  of  all  moving  bodies 
to  keep  in  a  straight  line  unless  drawn  from  it  by  some  other 
force  than  that  of  their  impulse,  and  also  that  bodies  moving 
in  circular  and  elliptical  orbits  tend  to  fly  off  from  the  centre 
unless  held  in  their  place  by  some  external  force.  As  the 
planets  were  moving  in  an  elliptical  orbit,  and  the  satellites 
about  their  central  bodies,  there  must  be  this  tendency  to 
fly  off  from  their  centres,  and  unless  they  deviated  from  this 
line  there  must  be  some  force  to  sustain  them  in  their  place. 
Gravitation,  therefore,  came  in  as  the  complementary  centrip- 
etal force  to  counterbalance  the  action  of  centrifugal  tenden- 
cies. The  existence  of  some  such  influence  was  presumed  in 
the  nature  of  the  case,  but  that  it  should  be  the  same  gravity 
that  pulled  the  apple  to  the  earth  was  not  proved  by  that  fact, 
and  hence  it  remained  a  probable  inference  of  greater  or  less 
degree,  according  as  the  nature  of  the  case  would  make  it. 
The  inductive  nature  of  the  inference  lies  precisely  in  this 
fact,  that  it  is  not  necessarily  conclusive,  but  only  that  the 
cause  inferred  is  adequate  to  the  effect,  and  that  the  circum- 
stances render  it  probable  that  the  supposition  is  true,  or  that 
it  has  more  in  its  favor  than  any  other  hypothesis.  The  mere 
possibility  of  some  other  conditions  to  the  same  effect  prevents 
the  case  from  being  proved  by  the  circumstances  which  occa- 
sioned the  inference.  But  it  will  increase  in  probability  with 
the  number  of  incidents  consistent  with  it,  or  which  it  aids  in 
making  intelligible.  In  the  case  before  us  the  probability  of 
gravitation  was  greatly  increased  by  the  mere  circumstance 
that  a  centripetal  force  was  needed,  and  the  conjecture  ex- 
tending to  the  moon  the  agency  which  caused  the  apple  to  fall 


INDUCTIVE  REASONING  305 

precisely  supplied  this  want.  In  getting  a  clear  notion  of 
what  the  inductive  inference  was,  however,  the  student  has 
only  to  remember  that  it  consisted  in  the  extension  to  the  va- 
rious planets  of  the  same  force  known  or  supposed  to  control 
the  movements  of  the  apple,  and  under  circumstances  which 
prevented  the  action  of  such  a  force  from  being  an  observed 
fact. 

Any  number  of  similar  illustrations  might  be  chosen.  For 
instance,  if  we  find  that  two  or  three  gases  are  compressed 
into  liquids  under  certain  degrees  of  temperature  and  press- 
ure, we  might  infer  the  same  of  other  gases  not  yet  so  com- 
pressed. The  conjunction  of  a  solar  eclipse  with  the  dark  of 
the  moon  would  suggest  the  inference  that  the  moon  was  the 
cause  of  the  phenomenon.  The  known  fact  that  intervening 
bodies  cast  a  shadow  is  the  basis  of  supposing  that  the  same 
effect  will  take  place  when  the  moon  is  between  the  earth  and 
the  sun.  Our  ignorance  of  the  exact  position  of  the  moon 
would  make  the  inference  merely  conjectural,  but  a  knowl- 
edge of  the  fact  that  its  latitude  and  longitude  corresponded 
exactly  with  the  position  of  the  sun  at  the  place  would  be 
equivalent  to  a  demonstration  of  the  inference,  or  would  make 
it  so  highly  probable  that  a  very  peculiar  combination  of  cir- 
cumstances would  be  required  to  weaken  it.  But  it  must  be 
noticed  that  the  circumstance  rather  verities  than  instigates 
the  inference,  and  becomes  the  basis  of  predictions  of  the 
eclipse.  Again,  the  peculiar  change  in  the  shape  of  sun  spots 
leads  to  the  inference  both  of  the  rotundity  and  the  axial  revo- 
lution of  the  sun.  They  move  across  the  visible  plane  of  the 
sun's  surface,  and  appear  elliptical  or  elongated  as  they  ap- 
proach the  edge  of  the  sun.  The  known  fact  of  such  elon- 
gation in  connection  with  spherical  bodies  would  suggest  sphe- 
ricity in  the  sun  in  connection  with  the  elongated  character  of 
the  spots,  and  the  conjunction  of  this  form  in  the  sun  spots 
with  their  motion  across  the  sun's  surface  suggests  axial  revo- 
lution according  to  well-known  facts.  And  again,  the  frequent 
conjunction  of  a  certain  kind  of  cloud  and  rainfalls  will  lead 

me  to  suppose  that  this  kind  of  cloud  is  a  cause,  and  that 
20 


30G  ELEMENT*  OF  LOGIC 

it  will  be  invariably  accompanied  with  rain.  To  Benjamin 
Franklin  certain  resemblances  between  lightning  and  electric- 
ity led  to  the  inference  that  they  would  resemble  each  other  in 
conduction  over  a  wire  and  in  charging  a  Leyden  jar.  The 
identification  of  the  two,  and  hence  the  verification  of  his  sup- 
position, was  the  result  of  an  experiment  suggested  by  this 
inference,  and  is  too  well  known  to  be  repeated. 

These  illustrations  suffice  to  indicate  the  nature  of  induc- 
tive inferences,  and  it  remains  to  show  the  syllogistic  form 
which  they  assume.  The  premises  consist  of  certain  facts 
and  assumptions,  and  the  conclusion,  of  something  wider  than 
the  known  facts  in  the  premises.  In  representing  them,  how- 
ever, we  cannot  follow  the  usual  order  of  the  Figures  in  the 
syllogism  without  too  much  preliminary  explanation.  We  take 
first,  therefore,  the  form  most  frequently  adopted. 

2d.  The  Form  of  the  Inductive  Syllogism. — The  usual 
form  of  inductive  reasoning  is  that  of  the  third  Figure.  My 
own  opinion,  however,  is  that  it  can  be  stated  in  all  the  Fig- 
ures. But  we  take  first  the  Third  Figure,  because  it  is  un- 
doubtedly the  one  in  which  the  inductive  inference  occurs 
most  frequently.     Thus  : 

Mars,  Venus,  etc.,  revolve  around  the  sun. 

Mars,  Venus,  etc.,  are  planets. 

Therefore  the  planets  revolve  around  the  sun. 

In  deductive  reasoning  we  have  learned  that  this  would  be  a 
case  of  illicit  process  of  the  minor  term,  because  it  is  undis- 
tributed in  the  minor  premise,  but  distributed  in  the  conclu- 
sion. But  we  must  remember  that,  although  the  inference  or 
reasoning,  deductively  considered,  is  false,  the  conclusion  may 
be  true  as  a  matter  of  fact.  Only  we  have  no  right  from  the 
nature  of  our  premises  to  assert  it.  This  indeterminate  nat- 
ure of  the  fact  leaves  it  open  to  conjecture,  hypothesis,  or  in- 
ductive reasoning  to  infer  that  all  the  planets  revolve  about 
the  sun,  because  two  or  three  of  them  do  ;  that  is.  that  the 
other  planets  being  like  the  known  planets  in  many  respects, 
will  prove  to  be  bike  them  in  this  other  characteristic.     It  is,  of 


INDUCTIVE  REASONING  307 

course,  not  a  necessary  inference  from  the  data  given,  but  only 
a  possible  or  probable  one,  as  the  case  may  be. 

The  matter  is  sometimes  put  in  the  following  manner  in  or- 
der to  avoid  the  appearance  of  a  deductive  fallacy : 

A,  B,  C  (magnets),  attract  iron. 
A,  B,  C  represent  all  magnets. 
Therefore  all  magnets  attract  iron. 

Here  the  inference  is  undoubtedly  wider  than  the  known 
facts.  But  having  used  the  phrase  "  rej>resent  all  magnets," 
we  distribute  the  term  magnets,  so  as  either  to  bring  the  case 
under  the  rules  of  the  quantification  of  the  predicate,  and, 
therefore,  deductive  reasoning,  or  we  create  one  of  those  in- 
tensive propositions  with  the  quantification  of  one  of  its  terms 
which  modifies  the  real  form  of  the  reasoning,  and  makes  it 
some  other  Figure  than  the  apparent  one,  as  has  been  ex- 
plained. Besides,  it  is  to  assume  the  conclusion,  if  we  thus 
express  the  case  in  the  premises,  and  so  far  from  having  our 
knowledge  increased  by  the  inference,  as  it  should  be  by  In- 
duction, it  remains  the  same  as  in  the  premises  in  such  cases. 
Hence  I  do  not  think  this  form  of  expression  conducive  to 
the  proper  representation  of  the  inductive  form  of  inference. 
We  must  not  assume  as  known  in  the  premises  what  we  are 
to  infer  in  the  conclusion.  Hence  the  better  form  for  show- 
ing the  nature  of  the  inductive  inference  is  such  as  the  follow- 
ing, where  nothing  is  assumed  in  the  premises  to  anticipate  or 
necessitate  the  conclusion. 

All  magnets  attract  iron. 

All  magnets  are  attracting  bodies. 

Therefore  all  attracting  bodies  attract  iron. 

We  may  know  the  premises  to  express  facts  and  yet  they  may 
not  include  the  conclusion  drawn  from  them.  But  in  this  in- 
stance the  agreement  or  connection  between  the  two  predi- 
cates and  the  same  subject  awakens  the  supposition  that  they, 
the  predicates,  are  as  essentially  connected  with  each  other. 
This  is,  of  course,  the  thing  to  be  proved.     But  it  is  possible 


308  ELEMENTS  OF  LOGIC 

in  the  nature  of  the  circumstances,  and  will  be  probable  ac- 
cording to  the  extent  of  our  knowledge  regarding  the  nature 
and  action  of  electricity.  It  may  even  reach  the  stage  of 
demonstration.  But  the  first  degree  is  that  of  possibility  or 
probability  suggested  by  certain  known  resemblances. 

In  the  second  Figure  the  reasoning  might  be  represented 
as  follows  : 

Magnets  attract  iron. 

Loadstones  attract  iron. 

Therefore  loadstones  are  magnets. 

In  deductive  reasoning  this  is  a  case  of  illicit  middle.  But 
for  the  same  reason  as  before  it  is  possible  that  the  two 
subjects  agree,  although  this  is  not  necessarily  the  case.  Their 
agreement  in  the  matter  of  attracting  iron  is  merely  the  con- 
ception of  a  single  effect  which  has  its  cause,  and  as  both  ob- 
jects produce  the  effect  we  may  reasonably  suppose  that  they 
are  identical  in  their  nature.  This  is  what  we  express  by  say- 
ing that  "  Loadstones  are  magnets." 

In  this  illustration  we  must  not  mistake  the  historical  man- 
ner in  which  magnets  and  loadstones  were  actually  identified 
for  the  inductive  inference  we  are  trying  to  illustrate.  His- 
torically, "magnets  "  was  only  another  name  for  "loadstones," 
and  we  came  afterward  to  apply  the  same  name  to  manufact- 
ured articles  having  the  same  qualities.  But  we  are  here 
supposing  a  mind  made  acquainted  with  the  two  things  inde- 
pendently of  each  other,  and  discovering  that  both  attracted 
iron  in  the  same  way,  and  hence  inferring  that  they  belonged 
to  the  same  class.  In  this  case,  then,  we  might  say  either  that 
the  loadstone  was  a  magnet,  or  the  magnet  a  loadstone.  We 
should  only  intend  by  it  that  one  of  them  was  either  possibly 
or  really  more  comprehensive  than  the  other. 

The  difficulty  with  the  second  Figure  in  the  inductive  syl- 
logism is  that  it  is  usually  impossible  to  connect  the  two 
subjects  by  the  copula  which  signifies  their  identity  in  a  class 
relation,  which  is  not  always  capable  of  being  expressed  by 
language  which   is   adjusted    to  observations   already  made. 


INDUCTIVE  REASONING  309 

There  is  a  reason  for  this  difficulty  which  will  appear  in  the 
sequel.  It  remains  to  see  whether  we  can  state  the  inductive 
syllogism  in  the  form  of  the  first  Figure. 

Meteors  are  followed  by  a  train  of  light 
Rapidly  moving  bodies  are  like  meteors  in  their  motion. 
Therefore,  rapidly  moving  bodies  are  like  meteors  in  being 
followed  by  a  train  of  light  (other  things  being  equal). 

The  conclusion  in  this  example  is  not  meant  to  indicate  an  ob- 
served fact,  but  for  that  very  reason  it  is  better  calculated  to 
represent  the  inductive  nature  of  the  inference.  It  is  not  a 
matter  of  experience  that  "  all  rapidly  moving  bodies  "  are  fol- 
lowed by  a  train  of  light,  and  yet  we  are  accustomed  to  ex- 
plain the  absence  of  this  effect  by  saying  that  we  mean  such 
bodies  as  have  the  velocity  of  meteors.  Our  argument  is  that 
if  they  had  as  swift  a  motion  as  meteors  they  would  display 
the  same  effects,  and  the  inference  that  they  would  do  so  is 
inductive,  in  so  far  as  it  supposes  true  of  all  kinds  of  matter 
what  is  observed  to  be  true  of  meteoric  matter  and  bodies.  It 
might  not  be  true  of  terrestrial  bodies,  because  of  their  pecu- 
liar nature  and  established  connection  with  the  earth.  But, 
on  the  other  hand,  the  effect  would  be  probable  in  proportion 
to  the  knowledge  we  have  of  the  effect  of  friction  from  the  air 
upon  rapidly  moving  matter  of  whatever  kind,  and  the  temper- 
ature necessary  to  make  it  luminous,  etc. 

In  regard  to  the  form  of  this  syllogism  it  will  be  noticed 
that  the  middle  term  is  quite  peculiar.  We  could  not  state 
that  the  subject  of  the  minor  premise  was  contained  in  the 
class  "  meteors,"  because  this  would  have  made  the  argument 
deductive,  while  it  is  our  object  either  to  "  prove  "  their  in- 
clusion in  that  class,  or  to  show  that  both  classes  belong  to  the 
same  general  kind  of  objects  on  the  ground  of  a  common 
quality  observed  and  a  common  quality  inferred,  which  shall  be 
as  essential  to  the  subject  of  the  conclusion  as  to  that  of  the  ma- 
jor premise.  Hence  we  must,  in  the  first  Figure,  state  our  mid- 
dle term  in  such  a  way  as  to  represent  an  identity  between  it  and 


310  ELEMENTS  OF  LOGIC 

the  minor  term  which  is  only  partial,  and  from  which  we  may  in- 
fer a  more  complete  identity  inductively.  Also  we  must  state  it 
so  that  it  represents  properly  the  extent  of  our  actual  knowledge 
about  the  two  terms,  so  that  the  conclusion  shall  contain  more 
than  our  actual  experience  of  the  facts.  In  the  minor  prem- 
ise, therefore,  we  cannot  say  that  the  subject  is-  the  predicate 
or  middle  term,  but  only  that  it  is  like  it  in  certain  particulars, 
and  then  we  can  infer  its  identity  with  the  major  term  so  as  to 
complete  the  conception  of  the  identity  between  the  two  classes 
of  objects.  Otherwise,  as  we  have  remarked,  the  argument 
would  be  purely  deductive.  It  will  be  interesting  to  remark 
that  this  is  the  valid  inductive  inference,  the  counterpart,  and 
correspondent  of  the  invalid  deductive  process  in  Ambiguous 
Middle. 

It  will  not  be  necessary  for  present  purposes  to  represent 
an  inductive  syllogism  in  the  fourth  Figure.  We  have  in- 
tended only  to  show  that  so  far  was  Induction  is  reasoning  it 
must  follow  the  same  form  as  Deduction,  and  that  it  differs 
from  Deduction  in  the  character  of  its  conclusion,  permitting 
us  to  go  beyond  the  premises  with  our  inferences,  although 
affording  nothing  but  a  probability  of  truth  instead  of  a  cer- 
tainty or  necessity.  But  there  is  a  peculiar  characteristic 
which  is  best  illustrated  in  the  example  representing  an  induc- 
tive syllogism  of  the  first  Figure,  and  which  is  our  reason  for 
having  given  it.  We  have  observed  the  peculiar  form  of  state- 
ment necessary  to  prevent  the  ease  from  being  a  deductive 
syllogism,  and  in  comparing  mathematical  with  ordinary  logical 
or  deductive  reasoning  we  remarked  that  the  former  was  purely 
quantitative  and  the  latter  both  quantitative  and  qualitative. 
We  have  now  still  further  to  observe  that  inductive  reasoning 
is  purely  qualitative.  Mathematical  reasoning  we  found  to  be 
quantitative  because  it  dealt  only  with  quantity,  and  was  not 
concerned  with  the  qualities  which  constitute  objects,  and  we 
chose  to  call  it  Traduction  because  its  data  could  be  trans- 
posed and  substituted  without  any  reference  to  the  Figure  of 
the  syllogism.  Induction,  on  the  other  hand,  has  nothing  to 
do  with   quantity,  but  solely  with  the  qualities  of  things.     It 


INDUCTIVE  REASONING  311 

reasons  from  quality  to  quality,  from  attribute  to  attribute. 
Thus  in  identifying  electricity  and  magnetism  we  argue  from 
certain  resemblances  of  effect  to  an  identity  of  cause,  although 
in  a  modified  form  in  each  instance.  We  do  not  have  any  special 
class  or  numerical  quantity  of  instances  under  which  the  minor 
premise  can  be  subsumed,  and  hence  we  argue  from  the  known 
qualities  in  one  or  more  things  to  the  existence  of  the  same 
quality  in  other  objects.  The  number  of  objects  specified  in  the 
premises  is  of  no  importance.  Each  premise  may  represent 
only  a  single  individual,  and  the  data  be  sufficient  for  an  in- 
ductive inference  of  some  kind.  Indeed,  when  class  terms  are 
employed  they  have  no  more  value  than  singular  .terms,  except 
for  strengthening  our  convictions  about  the  fixity  of  the  con- 
nection between  the  subject  and  the  predicate.  We  do  not  so 
much  use  them  for  class  or  general  terms  as  for  conceptions 
representing  the  presence  of  certain  qualities,  and  it  is  from  the 
peculiar  conjunction  of  certain  qualities  xoith  each  other  that  ive 
infer  a  similar  conjunction  where  it  has  not  yet  been  observed. 
Thus  we  are  not  arguing  from  a  class  to  a  sub-class  or  indi- 
vidual under  it,  on  the  ground  of  numerical  inclusion  in  the 
larger,  but  from  a  quality  or  qualities  observed  to  the  same 
things  unobserved  in  other  objects,  or  in  a  modified  form  con- 
cealing their  identity.  Thus  Franklin,  in  arguing  to  the  iden- 
tity of  lightning  and  electricity,  reasoned  from  certain  known 
phenomena,  the  production  of  sparks  by  an  electric  current,  to 
the  production  of  the  same  by  lightning,  if  the  proper  condi- 
tions were  satisfied  and  his  flying  of  the  kite  was  only  a  veri- 
fication of  the  inference.  The  number  of  instances  in  which 
he  had  observed  electrical  phenomena  and  lightning  had  noth- 
ing to  do  with  his  reasoning,  but  only  in  strengthening  his 
conviction  regarding  the  permanence  or  uniformity  of  the  re- 
semblance between  the  two  sets  of  phenomena.  The  logical 
process  was  then  not  concerned  with  the  mathematical  aspect 
of  his  conceptions,  but  only  with  their  qualitative  import ; 
that  is,  with  the  resemblances  between  phenomena.  A  veri- 
fied induction  may  enlarge  the  area  or  extension  of  general 
concepts,  but  it  is  not  done  by  any  logical  stress  upon  the 


312  ELEMENTS  OF  LOGIC 

mathematical  import  of  the  conceptions  employed.  It  is  done 
only  by  observing,  or  inferring,  and  then  verifying  a  resem- 
blance of  qualities.  After  the  general  conception  has  been 
formed,  we  may  use  its  quantitative  import  as  a  basis  for  as- 
sured deductive  reasoning,  but  only  the  qualities  of  objects 
and  their  connections  can  be  the  basis  of  induction,  and  hence 
we  feel  justified  in  calling  it  purely  qualitative  reasoning. 

We  must  be  on  our  guard,  however,  about  the  use  of  the 
terms  quantitative  and  qualitative  in  describing  these  several 
kinds  of  reasoning,  because  they  are  used  here  in  a  somewhat 
new  and  modified  sense.  Jevons,  for  instance,  and  most  writers 
on  Logic,  speak  of  "quantitative"  reasoning  when  they  mean 
reasoning  with  extensive  propositions,  and  "qualitative"  rea- 
soning when  they  mean  reasoning  with  intensive  propositions. 
The  meaning  I  attach  to  the  terms  rather  includes  this  than 
differs  from  it  essentially,  because  I  have  already  laid  it  down 
as  a  fact  that  all  extensive  propositions  have  a  corresponding 
intensive  import,  and  intensive  propositions  an  extensive  im- 
port ;  so  that  they  may  be  conceived  either  quantitatively  or 
qualitatively.  But  by  quantitative  reasoning  I  mean  the  com- 
parison of  terms  on  the  ground  of  quantitative  identity,  or 
quantitative  relations  of  whole  and  part,  and  by  qualitative 
reasoning,  the  comparison  of  terms  on  the  ground  of  a  qualita- 
tive identity,  which  never  permits  of  additions  to  form  quanti- 
tative wholes  ;  and  hence  quantitative  or  mathematical  reason- 
ing will  deal  with  conceptions  in  purely  numerical  relations  ; 
qualitative  or  inductive  reasoning  only  with  the  qualitative  re- 
semblances and  connections  of  phenomena.  If,  then,  we  can 
apply  the  term  Traduction  to  mathematical  reasoning,  we  may 
indicate  briefly  the  characteristics  of  the  three  forms  of  infer- 
ence. Traduction  is  purely  quantitative,  the  most  assured  in 
its  certitude  and  the  most  free  from  fallacy.  Induction  is 
purely  qualitative,  the  least  assured  in  its  conclusions  and  the 
least  exempt  from  mistakes  of  inference.  Deduction  is  both 
quantitative  and  qualitative,  and  so  combines  the  characteris- 
tics of  both  Traduction  and  Induction.  It  is  assured  in  its 
conclusions   precisely   in   proportion  as  the  reasoning  turns 


INDUCTIVE  REASONING  313 

upon  the  quantitative  import  of  its  conceptions  and  proposi- 
tions, and  it  is  exposed  to  fallacies  precisely  in  proportion  to 
the  lack  of  coincidence  between  the  group  of  qualities  consti- 
tuting the  individuals  in  a  class  of  objects  and  the  mathemati- 
cal import  of  the  class  term  denominating  them.  That  is,  if 
the  same  term  has  a  wider  mathematical  application  than  the 
group  of  qualities  it  may  denote  there  is  great  liability  to  fal- 
lacies of  all  kinds  iu  deductive  reasoning,  and  hence  when  the 
question  turns  upon  the  qualitative  import  of  terms  we  see 
the  probability,  or  at  least  the  possibility,  of  error  increas- 
ing. 

3d.  Kinds  of  Inductive  Inference  and  its  Principles, 
— The  inductive  conclusion  does  not  always  take  the  exact 
form  which  has  been  given  it  in  our  illustrations.  "We  had  in 
view  such  a  statement  of  the  process  as  would  give  a  clear  idea 
of  what  it  was  iu  its  essential  characteristics,  and  so  to  sepa- 
rate it  from  that  class  of  conclusions  which  were  the  result  of 
verification  or  of  a  surreptitious  introduction  into  the  prem- 
ises of  matter  which  ought  first  to  appear  in  the  inference. 
But  there  is  some  confusion  about  what  matters  of  fact  should 
belong  to  an  inductive  inference,  and  this  is  generally  caused 
by  the  mode  of  representing  its  syllogistic  form.  This  con- 
fusion, however,  is  not  one  with  which  logicians  have  dealt,  or 
about  which  there  has  been  any  controversy.  It  is  rather  the 
unconscious  classification  under  the  head  of  inductive  reason- 
ing of  two  or  more  kinds  of  general  conceptions,  beliefs,  or 
principles,  which  are  suggested  by  individual  facts  much  nar- 
rower than  the  truths  they  give  rise  to,  that  has  caused  the 
confusion.  They  are  spoken  of  as  inductive  inferences  with- 
out taking  into  account  either  the  complications  and  condi- 
tions that  make  some  of  them  appear  quite  different  from 
others,  or  the  various  degrees  of  certitude  that  attach  to  them, 
making  some  of  them  quite  assured  and  others  quite  conjec- 
tural. 

We  spoke  of  "the  surreptitious  introduction  into  the  prem- 
ises of  matter  which  ought  first  to  appear  in  the  conclusion." 
if  the  inference  were  to  be  truly  inductive,  as  characteristic  of 


314  ELEMENTS  OF  LOGIC 

some  representations  of  the  process.  Hence  it  is  common  to 
illustrate  inductive  reasoning  in  the  following  manner  : 

The  Earth  is  molten  under  Vesuvius. 

Vesuvius  fairly  represents  the  interior  of  the  Earth. 

Therefore  the  interior  of  the  Earth  is  molten. 

The  only  appearance  of  induction  in  this  instance  lies  in  the 
fact  that  the  major  premise  is  a  particular  circumstance,  and 
the  conclusion  another  supposed  to  be  suggested  by  it.  It 
also  appears  to  be  a  syllogism  of  the  fourth  Figure,  but  when 
stated  in  another  form  it  appears  to  be  of  the  third  Figure. 
Thus, 

The  Earth  under  Vesuvius  is  molten. 

The  Earth  under  Vesuvius  fairly  represents  the  interior  of 

the  Earth. 
Therefore  the  interior  of  the  Earth  is  molten. 

But  even  this  is  not  the  completely  correct  form  in  which 
the  logical  relation  of  the  terms  is  actually  thought,  because 
the  only  conclusion,  inductive  or  deductive,  which  can  be 
drawn  from  these  premises  is,  "That  which  fairly  represents 
the  interior  of  the  Earth  is  molten."  The  minor  premise,  as 
we  have  indicated  before  in  propositions  of  this  kind,  is  an  in- 
tensive judgment  which  requires  reduction  to  express  the  true 
relation  of  the  terms  in  it,  and  hence  it  would  be,  "  The  inte- 
rior of  the  Earth  is  like  that  under  Vesuvius."  But  this  makes 
the  case  one  of  AAA  in  the  first  Figure,  and  also  a  deductive 
syllogism,  unless  we  so  state  the  resemblance  as  not  to  indi- 
cate that  it  is  complete,  but  only  in  a  matter  that  supports  the 
probability  of  a  conjunction  with  the  predicate  "molten."  In 
stating  the  likeness  in  the  jDremises  as  we  do,  we  practically 
assume  in  them  all  that  is  found  in  the  conclusion,  and  so 
many  of  the  cases  which  are  taken  to  represent  inductive  are 
simply  instances  of  deductive  reasoning.  The  appearance  of 
induction  comes  only  in  the  admitted  iwobabUity  of  the  conclu- 


INDUCTIVE  REASONING  315 

sion.  But  this  characteristic  is  due  solely  to  the  probability 
of  one  or  perhaps  both  of  the  premises.  If  the  premises,  stated 
in  the  manner  of  the  above  illustration,  be  proved  facts,  the 
conclusion  is  a  necessary  one,  and  not  probable  at  all.  The 
probability  of  an  inductive  inference  is  one  which  is  the  result 
of  an  inference  from  established  facts,  and  so  is  due  to  going 
beyond  the  premises  under  the  stimulus  of  certain  general  prin- 
ciples which  may  be  called  the  Principles  of  Induction.  These 
will  be  considered  in  a  moment.  But  when  we  say  in  the  mi- 
nor premise  of  an  inductive  syllogism  that  certain  things  "  rep- 
resent "  the  subject  of  the  major,  or  "  are  like "  it,  without 
qualification,  we  assume  in  it  what  ought  to  be  an  inference  in 
the  conclusion  from  admitted  or  supposed  facts  which  do  not 
include  the  inference.  The  probability  of  the  conclusion  must 
not  be  borrowed  from  that  of  the  premise,  but  from  the  prin- 
ciples which  justify  inferences  beyond  the  actual  data  given, 
otherwise  we  should  practically  be  guilty  of  a  petitio  principii. 
Thus  if  we  were  asked  to  give  inductive  proof  for  the  prop- 
osition that  "  the  interior  of  the  Earth  is  molten,"  we  should 
state  the  syllogism  as  follows : 

The  Earth  immediately  under  Vesuvius  is  molten. 

The  interior  of  the  Earth  and  that  immediately  under  Vesu- 
vius resemble  those  conditions  in  which  openings  are 
outlets  of  what  is  contained  deeper  within. 

Therefore  the  interior  of  the  Earth  resembles  that  immedi- 
ately under  Vesuvius  in  being  molten. 

Here  are  two  known  facts  with  such  resemblances  as  inevi- 
tably suggest  a  further  resemblance,  that  of  a  quality  which  is 
observed  and  known  in  one  of  the  objects,  but  unobserved  in 
the  other.  It  is  also  to  be  noted  in  this  case  that  the  middle 
term  is  peculiar.  There  is  no  such  identity  or  inclusion  as  in 
Deduction.  It  is  only  partial,  and  this  is  the  true  characteris- 
tic of  inductive  reasoning.  The  known  facts  of  the  middle 
term  nmst  represent  partial  identity  and  resemblances,  and  the 
inference  must  contain  the  total  identity.     It  is  this  feature  of 


316  ELEMENTS  OF  LOCH' 

the  inductive  syllogism  or  reasoning  ;  namely,  the  purely  qual- 
itative, and  not  quantitative,  nature  of  the  comparison,  that 
lends  support  to  the  supposition  that  a  truly  formal  expres- 
sion of  the  process  is  impossible.  It  is  certainly  very  difficult 
in  many  cases.  But  aside  from  these  cpuestions  the  important 
feature  of  the  inductive  proof  is  the  statement  of  certain 
known  and  partially  agreeing  facts.  They  may  wholly  agree, 
or  they  may  only  have  a  partial  agreement,  but  which  of  the 
two  it  really  is  must  be  determined  by  the  verification  or  dis- 
proof of  the  inductive  inference.  All  that  we  are  presumed  to 
know  in  the  premises  is  their  partial  agreement  or  actual  con- 
nection, and  we  are  to  infer  from  them  under  the  circum- 
stances their  total  agreement  or  their  necessary  connection. 
We  are  not  to  assume  any  data  in  the  premises  which  should 
first  appear  in  the  conclusion. 

It  has  been  necessary  to  dwell  upon  this  point  in  order  to 
distinguish  clearly  between  deductive  probability  and  induc- 
tive probability,  as  a  condition  of  fixing  upon  truly  inductive 
inferences.  It  has  been  too  common  to  confuse  the  proba- 
bility of  a  conclusion  with  its  inductive  nature,  and  we  need 
to  know  exactly  the  marks  which  make  the  inference  the  one 
or  the  other.  In  deductive  reasoning,  if  one  or  both  of  the 
premises  be  probable  truths,  the  conclusion  will  be  probable, 
but  it  will  necessarily  follow  as  an  inference  from  the  supposed 
data.  But  in  inductive  reasoning  this  is  not  the  case.  The  in- 
ference is  not  a  necessary  one,  although  it  is  always  a  prob- 
able one.  Its  probability  is  independent  of  the  question 
whether  the  premises  are  positively  known  facts  or  only  likely 
assumptions.  It  comes  from  the  peculiar  nature  of  inductive 
principles  in  connection  with  its  extension  beyond  the  data  of 
the  premises.  The  difference,  therefore,  in  this  respect,  be- 
tween deductive  and  inductive  inferences  is  simply  this  :  In 
deduction  the  probability  of  the  conclusion  is  purely  material ; 
that  is,  determined  by  the  material  character  of  the  premises 
in  this  respect.  In  induction  the  probability  is  both  for m al- 
and material.  But  it  is  not  due  to  that  characteristic  in  the 
premises.     It  is  due  to  the  manner  in  which  the  inference  is 


INDUCTIVE  REASONING  317 

drawn,  and  this  is  the  introduction  of  new  matter  in  the  con- 
clusion. Hence  inductive  inferences  are  not  determined  by 
the  mark  of  probability  alone,  but  by  probability  plus  an  incre- 
ment of  knowledge,  or  conceptions  not  necessarily  involved  in  the 
j) remises.  This  maxim  must  be  constantly  kept  in  mind  or  the 
student  will  confuse  with  each  other  two  entirely  distinct 
kinds  of  reasoning. 

It  will  be  important  to  examine  this  liability  to  confusion 
because  of  its  bearing  upon  the  general  theory  of  inductive 
reasoning.  There  are,  at  least,  three  conceptions  of  the  pro- 
cess, which  are  not  necessarily  conflicting,  but  will  be  so 
upon  the  supposition  that  each  one  is  exclusive  of  the  other. 
They  are  : 

(1.)  That  the  inductive  inference  is  from  the  particular  to 
the  particular,  or  from  one  or  more  individual  cases  to  an- 
other. 

(2.)  That  it  is  from  the  particular  to  the  universal,  or  an  in- 
ference which  forms  a  generalization  from  a  particular  instance 
or  instances. 

(3.)  That  it  is  an  inference  from  actual  to  necessary  connec- 
tion, or  from  uniformities  of  coexistence  and  sequence  to 
causal  relations. 

It  can  be  shown  that  each  of  these  forms  of  inference  ulti- 
mately result  in  generalization,  and  this  is  probably  the  reason 
that  John  Stuart  Mill  identifies  inductive  reasoning  with  gen- 
eralizations from  particular  cases.  But  there  are  some  general- 
izations which  only  seem  to  be  from  particular  instances,  and 
hence  may  be  confused  with  inductive  inferences,  merely  be- 
cause the  data  we  start  with  happen  to  be  individual  facts  and 
the  conclusion  includes  more  than  these,  and  perhaps  all  sim- 
ilar facts,  when  in  the  meantime  a  deductive  process  has  been 
surreptitiously  but  unconsciously  introduced.  This  is  the  case 
when  a  probable  conclusion  in  a  deductive  syllogism  is  based 
upon  an  inductive  conclusion  as  one  of  the  premises,  or  when 
our  generalization  simply  states  explicitly  what  was  implicitly 
involved  in  the  conceptions  arrived  at  by  induction.  Let  us 
examine  the  application  of   these   principles  in  the  example 


31 S  ELEMENTS  OF  LOGIC 

which   Mr.  Fowler  presents   as   an   illustration   of  inductive 
reasoning,  as  follows : 

(a)  I  observe  that  these  two  bodies  (though  of  unequal 
weight)  reach  the  bottom  of  the  receiver  at  the  same  moment. 

(b)  This  fact  must  be  due  to  some  cause  or  combination  of 
causes  (Law  of  Universal  Causation). 

(c)  The  only  cause  operating  in  this  instance  is  the  action 
of  gravity. 

(d)  Therefore  the  fact  that  these  two  bodies  reach  the 
bottom  of  the  receiver  at  the  same  moment  is  due  to  the 
action  of  gravity  operating  alone. 

(e)  But  whenever  the  same  cause  or  combination  of  causes 
is  in  operation,  and  that  only,  the  same  effect  will  invariably 
follow  (Law  of  Uniformity  of  Nature). 

(/)  Therefore  whenever  these  two  bodies,  or  any  other  two 
or  more  bodies  (even  though  of  unequal  weight)  are  subject  to 
the  action  of  gravity  alone,  they  will  reach  the  bottom  of  the 
receiver  at  the  same  moment,  or,  in  other  words,  will  fall  in 
equal  times. 

The  first  remark  here  is  that  we  have  in  this  instance  two 
syllogisms,  one  ending  with  proposition  (d)  and  the  other 
with  proposition  if).  The  first  can  be  shown  to  be  purely  de- 
ductive, and  the  second  to  contain  both  a  deductive  and  an 
inductive  conclusion.  Whether  Mr.  Fowler  intended  that 
proposition  (d)  should  be  an  inductive  inference  I  cannot  say. 
But  it  is  certain  that  the  only  appearance  of  such  a  character 
in  the  inference  is  the  nature  of  proposition  (a),  which  merely 
states  a  fact  or  phenomenon,  and  of  proposition  (d),  which  in- 
cludes the  cause  of  the  phenomenon.  In  the  meantime,  how- 
ever, if  we  observe  closely,  all  the  conditions  for  making  the 
conclusion  deductive  are  introduced  into  propositions  (6)  and 
(c).  Propositions  (b),  (c),  and  (d)  form  a  complete  deductive 
syllogism,  with  (c)  as  the  major  and  (b)  as  the  minor  premise  ; 
or  the  mode  of  statement  can  so  be  altered  as  to  make  (b)  the 
major  and  (c)  the  minor  premise.  Let  us  see  how  this  is  the 
case. 

Proposition   (6)   is  not  an  inductive  inference,  nor  an  ob- 


INDUCTIVE  REASONING  319 

served  fact,  although  it  contains  new  data  not  found  in  jDropo- 
sition  (a).  It  is  a  deductive  inference  from  the  assumption 
that  all  phenomena  must  have  a  cause,  and  the  observed  fact 
of  the  two  falling  bodies,  which  is  taken  to  be  a  phenomenon. 
Now  proposition  (b)  becomes  a  premise  for  another  syllogism, 
and  it  is  assumed  again  that  the  action  of  gravity  is  the  only 
operating  cause  in  the  case,  from  which  it  immediately  and 
necessarily  follows  that  this  instance  of  falling  bodies  has 
gravity  for  its  cause.  The  causal  influence  of  gravity,  instead 
of  being  directly  inferred  from  the  observation  of  a  fact,  is  as- 
sumed in  the  premise,  and  so  makes  the  conclusion  deductive. 
The  only  proper  way  to  state  the  inductive  inference  in 
such  cases  is  as  follows  : 

I  observe  that  two   bodies  of    unequal  weight  fall  through 

equal  spaces  in  equal  times. 
I  observe  that  the  action  of  gravity  is  isolated  at  the  same 

time. 
Therefore  the  isolated  action  of  gravity  is  the  cause  of  the 

phenomenon  ;  that  is,  of  the  falling  through  equal  spaces 

in  equal  times. 

In  this  form  of  statement  we  merely  assert  the  observed  coinci- 
dence of  two  facts,  the  falling  of  the  two  bodies  and  the  isola- 
tion of  gravity,  and  from  the  coincidence  we  infer  the  cause, 
or  from  their  actual  connection  we  infer  their  necessary  con- 
nection. This  is  not  involved  in  the  data  known,  although 
we  may  have  some  foreign  reason  for  making  the  inference. 
But  to  assume  that  gravity  is  a  cause  in  the  case,  and  more 
especially  that  it  is  the  only  cause,  is  to  introduce  in  the 
premises  what  we  find  in  proposition  (d),  and  so  make  the  in- 
ference deductive. 

In  the  second  syllogism  the  inferences  seem  to  be  some- 
what different.  They  are  that  the  same  two  bodies  will  al- 
ways fall  in  the  same  manner  under  the  same  conditions,  and 
that  all  other  bodies  will  do  so.  Here  we  have  two  extensions 
beyond  the  observed  fact.     But  we  must  deal  with  only  one 


320  ELEMENTS  OE  LOGIC1 

of  them  at  a  time.  We  take  the  first,  that  these  two  bodies 
will  always  fall  through  equal  spaces  in  equal  times,  and  con- 
sider it  in  relation  to  the  premises  from  Avhich  it  is  drawn. 

As  before,  we  have  a  deductive  instance  from  the  very  fact 
that  the  universality  of  the  case  is  assumed  in  proposition  (e). 
The  universal^  of  the  connection  must  be  suspected  or 
drawn  from  individual  cases  in  order  to  be  inductive.  Propo- 
sition (e)  really  states  what  should  be  the  inference,  and  since 
it  states  it  as  a  premise  the  conclusion  only  deduces  it  in  an- 
other form,  but  does  not  induce  it.  That  the  same  effect  will 
invariably  follow  the  operation  of  the  same  cause,  and  that 
alone,  is  the  truth  to  be  inferred  and  proved.  But  if  it  be 
stated  in  one  of  the  premises  its  presence  in  the  conclusion 
cannot  be  the  result  of  an  inductive  inference.  To  make  its 
universality  in  time  an  inductive  act  of  reasoning,  we  should 
compare  the  same  effects  with  each  other  in  several  different 
times,  and  then  from  the  observed  fact  that  a  difference  of 
time  had  exercised  no  influence  upon  the  result,  infer  that  this 
incident  alone  never  would  do  so,  and  that  with  the  changes 
of  time,  the  effect  would  always  be  the  same.  But  a  grave 
doubt  may  exist  about  the  inductive  nature  of  this  inference, 
because  to  infer  that  a  certain  event  will  always  happen  under 
the  same  conditions,  if  we  have  reason  to  believe  that  those 
conditions  are  its  cause,  or  necessarily  connected  with  it,  may 
well  be  claimed  to  be  an  a  priori  or  deductive  inference  on  the 
ground  that  the  universality  of  the  causal  connection  is  im- 
plied in  the  fact  of  its  existence  at  all  in  a  particular  case  and 
that  difference  of  time  is  always  implied  in  universality.  If 
this  be  so,  then  to  infer  that  an  event  will  always  occur  under 
the  same  circumstances  that  were  once  its  cause,  is  only  to 
state  explicitly  what  is  implicitly  involved  in  the  idea  of  their 
being  the  cause  in  the  first  place. 

A  complete  theory  of  Induction  would  require  us  to  dis- 
cuss this  question  at  length.  But  in  an  elementary  treatise  we 
cannot  be  expectel  to  do  so,  and  we  must  therefore  be  con- 
tent to  announce  the  conclusion  we  have  adopted  for  such 
cases.      It  is  that  all  such  generalizations   as  we   have  men- 


INDUCTIVE  REASONING  321 

tioned,  namely,  the  inference  that  a  given  event  will  always 
happen  under  the  same  circumstances,  are  deductive  when 
they  are  from  a  causal  connection  to  its  universality,  but  in- 
ductive when  we  reason  from  coincidences  and  sequences  to 
causes.  To  infer  a  causal  connection  in  a  single  instance  of 
observed  coincidences  is  of  the  nature  of  a  universal  gen- 
eralization, not  because  it  is  inferred  from  a  single  incident, 
but  because  the  universality  of  the  causal  relation  is  involved 
in  the  idea  of  its  necessity,  and  hence  no  new  conception  is  in- 
volved in  the  generalization.  But  to  infer  the  causal  connec- 
tion from  the  mere  fact  of  actual  connection  is  different,  and 
so  is  inductive.  Consequently  Mr.  Fowler's  first  inference  is 
deductive  because  it  only  states  explicitly  what  is  implied  in 
the  idea  of  necessary  connection  in  the  observed  case. 

These  observations  show  how  we  ax'e  to  dispose  of  the 
second  inference  in  proposition  (f)  of  Mr.  Fowler's  illustration. 
We  find  there  that  he  has  argued  from  the  instance  of  two 
bodies  of  unequal  weight  to  all  bodies  of  equal  or  unequal 
weight.  Inequality  of  weight  is  shown  by  the  observed  case 
not  to  be  a  material  circumstance,  but  for  all  that  we  know 
about  them  the  different  qualities  of  other  bodies  may  be 
precisely  the  circumstances  which  will  prevent  the  formation 
of  a  true  or  perfect  middle  term  for  the  syllogism,  and  pre- 
vent the  expected  effect  in  the  case  of  experiment.  These  dif- 
ferences are  not  included  in  the  premises,  and  hence  can  only 
be  included  with  a  certain  degree  of  probability  in  the  con- 
clusion, according  as  the  principles  of  induction  determine  it. 
For  this  reason,  therefore,  we  agree  that  the  inference  from 
the  specified  case  to  the  idea  that  all  bodies  will  fall  through 
equal  space  in  equal  times,  is  inductive,  because  it  is  a  gener- 
alization involving  an  increment  in  the  conclusion  not  contained 
in  the  premises.  To  suppose  it  true  of  all  bodies  is  to  infer 
it  of  other  conditions  than  those  in  which  it  was  first  in- 
ferred. 

We  are  now  prepared  to  examine  the  kinds  of  inductive  in- 
ference. They  may  be  divided,  first,  into  two  species  (1)  the 
statical  and  (2)  the  dynamical  inference.  By  a  statical  induc- 
21 


322  ELEMENTS  OF  LOGIC 

tive  inference  we  mean  an  inference  to  the  actual  existence  of 
certain  qualities,  coincidences,  or  sequences,  from  their  obser- 
vation in  given  cases.  This  class  we  subdivide  into  two  subor- 
dinate species  determined  by  the  diffi  rent  characteristics  men- 
tioned in  the  definition.  There  arc  first,  those  which  infer  r<  - 
semblances  or  differences,  or  the  existence  of  certain  qualities 
from  the  partial  resemblances  and  differences  of  observed 
cases.  The  second  class  consists  of  those  which  infer  the 
repeated  or  continued  existence  of  observed  coincidences  and 
sequences.  The  two  classes  often  coincide,  so  that  it  may  be 
difficult  to  decide  whether  the  inference  is  to  resemblances 
and  differences,  or  coincidences  and  sequences.  But  the  one 
or  the  other  aspect  predominates  frequently  enough  to  use 
the  difference  for  distinguishing  different  forms  of  the  same 
kind  of  reasoning.  By  a  dynamical  inductive  inference  we 
mean  an  inference  from  actual  to  necessary  connections.  The 
observed  incidents  may  be  of  coincidences,  or  sequences,  and 
the  inference  in  this  case  must  not  be  to  the  mere  probability 
of  their  recurrence,  but  to  the  probability  that  one  of  the  cir- 
cumstances is  the  cause  of  the  other.  Whenever  we  infer, 
therefore,  a  particular  cause  from  the  mere  occurrence  of  a 
phenomenon,  we  are  reasoning  inductively.  Reasoning  to  the 
existence  of  resembling  qualities  may  be  called  attributive  in- 
ductive inference  ;  to  coincidences  and  sequences,  may  be 
called  connect  ire  inductive  inference;  and  to  causes  may  be 
called  causal  inductive  inference.  The  causal  inference,  how- 
ever, may  be  implied  along  with  the  others,  and  hence  its 
distinction  from  them  may  not  be  absolute.  But  it  is  often 
so  prominent  a  feature  of  the  inference  as  to  serve  for  a  cri- 
terion of  its  nature.  In  general,  the  terms  statical  and  dy- 
namical comprehend  the  two  characteristics  determining 
inductive  inferences,  namely,  reasoning  to  unobserved  facts, 
whether  qualitative  or  connective,  and  the  causal  nexus.  The 
following  outline  summarizes  results  : 

Inductive    j  Statical  I  Attributive  =  Unobserved  identity  and  differences. 
Inferences  1  '  '-'onnective  =  Recurrent  coincidences  and  sequences. 

(  Dynamical  =  Causal  =  Necessary  connections. 


INDUCTIVE  REASONING  323 

Another  classification  of  inductive  inferences  can  be  made 
upon  the  basis  of  the  kinds  of  generalization  involved  in  them. 
Directly  or  indirectly,  primarily  or  ultimately,  all  inductive 
inferences  result  in  a  generalization,  explicit  or  implicit,  and 
hence  we  may  be  able  to  distinguish  them  by  this  characteris- 
tic. But  it  is  not  all  generalizations  that  are  inductive  infer- 
ences. We  have  seen  that  those  of  "  perfect  induction  "  are 
not  cases  of  reasoning  at  all ;  so  also  with  the  universalizing  of 
the  causal  connection.  Hence  we  must  assign  the  characteris- 
tics which  mark  the  generalizations  of  inductive  inferences. 
They  are  (1)  inferences  from  one  particular  case  to  another,  or 
from  species  to  species.  This  results  in  forming  or  implying 
a  larger  class,  comprehending  both  particulars,  and  so  identi- 
fying them  in  a  higher  genus.  Thus  if  I  infer  the  qualitative 
identity  of  magnets  and  loadstones,  of  electricity  and  magnet- 
ism, of  potassium  and  metals,  from  resemblances  of  qualities 
or  of  effects  produced  by  them,  I  am  forming  a  higher  genus, 
or  as  in  the  case  of  potassium  and  the  metals,  widening  the 
last  class.  In  chemistry,  biology,  and  zoology  this  inference  is 
very  frequent.  It  is  very  common  in  the  classification  of  ani- 
mals, and  especially  in  investigations  involving  the  doctrine  of 
evolution.  There  are,  then,  (2)  inferences  from  part  to  whole, 
from  particular  to  universal,  or  from  the  individual  to  the 
class,  from  the  species  to  the  genus.  This  form,  however,  needs 
explanation.  It  might  be  better  to  define  it  as  an  inference 
from  accident  to  essence,  or  from  differentia  to  conferentia, 
only  we  must  not  assume  that  the  basis  of  the  inference  is 
known  to  be  an  accident  or  a  difference.  It  is  merely  an  ob- 
served fact  that  a  certain  quality  is  present,  and  when  we  in- 
fer that  it  will  be  found  to  characterize  the  class  we  infer  that 
it  will  be  a  common  quality,  and  we  thus  pass  from  the  rela- 
tion of  accident  or  difference  to  that  of  essence  or  conferentia. 
Thus  if  we  observe  that  very  frequently  ruminants  are  horned, 
and  infer  that  all  ruminants  are  homed,  we  do  not  extend  the 
genus  as  a  class  term,  but  we  merely  increase  the  number  of 
qualities  to  be  taken  as  the  essence  or  conferentia  of  the  class  ; 
we  infer  that  what  might  be  a  mere  accident  is  probably  an  es- 


324  ELEMENTS  OF  LOGIC 

scntial  property.  Or  if  we  infer  from  the  conjunction  of  a 
single  or  several  east  winds  with  rainfall  that  all  cast  winds 
must  be  so  characterized,  we  infer  that  rainfall  is  an  essential, 
not  an  accidental,  accompaniment  of  east  winds.  AVe  do  not 
widen  the  genus,  but  we  increase  the  conferenti.i . 

It  is  important  to  remark  the  difference  between  these  two 
classes  of  generalization.  The  firsi  class  widens  the  genus,  ami 
the  second  does  nut.  The  first  does  not  change  the  conceived 
character  of  the  quality  or  relation,  but  only  its  extent,  and  the 
second  does  so  change  it.  A  further  important  difference  is, 
that  in  the  fust  class  the  resemblances  between  the  particulars 
are  actually  observed,  and  it  is  the  identity  of  the  cause  or 
quality  producing  them,  when  that  identity  of  kind  is  other- 
wise not  apparent,  that  constitutes  the  object  of  the  inference. 
In  the  second  class,  the  whole  genus  is  known,  not  inferred, 
from  the  individual  or  species  ;  but  it  is  merely  a  sufficient 
number  of  common  qualities,  or  conferentiae  to  determine  the 
group  of  individuals  named.  The  resemblances,  however,  are 
not  observed,  but  inferred.  The  coincidence  of  the  essentia 
with  a  quality  which  might  be  an  accident  is  observed,  and 
then  its  universal  coincidence  with  the  essentia  inferred,  sup- 
posing that  it  is  a  general  instead  of  an  accidental  characteris- 
tic. This  is  why  we  may  be  said  in  some  cases  to  pass  from 
the  individual  to  the  universal,  or  from  the  species  to  the 
genus.  It  is  only  when  the  genus  is  ab*eady  known  by  a 
definite  essentia  or  conferentia,  that  this  inference  can  be 
made.  Otherwise  we  should  have  to  depend  upon  "perfect 
induction,"  or  the  addition  of  individual  instances  in  order  to 
form  a  generalization.  It  may  be  that  observation  forms  the 
first  generalization  ;  but  afterward  the  process  may  be  materi- 
ally aided  by  the  inference  we  have  just  discussed. 

The  two  generalizations  we  have  just  considered  are  stati- 
cal. When  we  come  to  the  third  and  dynamical  we  have  to 
be  cautious  in  our  judgment  of  their  inductive  nature,  as 
we  have  already  observed  in  our  examination  of  Mr.  Fowler's 
example.  But  there  is,  nevertheless,  a  third  class  of  generab- 
zations  that  are  inductive.     They  are  (3)  inferences  from  act- 


INDUCTIVE  REASONING  325 

ual  to  necessary  and  universal,  or  causal  connections.  The 
basis  of  these  inferences  must  always  be  the  coincidences  or 
sequences  of  two  or  more  phenomena,  and  the  increment  sup- 
plied by  the  inference  must  be  either  the  causal  nexus  or  the 
universality  of  the  coincidence  and  sequence,  or  both.  But 
they  are  very  much  complicated.  "We  sometimes  directly  in- 
fer that  a  given  phenomenon  is  the  cause  of  another,  from  the 
fact  of  their  connection,  and  then  we  infer  that  it  will  always 
be  so.  But  in  the  last  instance  we  may  imply  that  the  condi- 
tions are  absolutely  the  same  for  all  cases,  or  that  if  they  are, 
the  events  will  always  happen  in  this  way.  If  this  be  our 
assumption  the  inference  is  due  to  a  subsumption  under  the 
universal  law  of  causation,  and  becomes  deductive,  or  merely 
an  explicit  statement  of  what  is  involved  in  the  idea  of  causa- 
tion, as  already  explained.  Ou  the  other  hand,  if  we  infer 
that  a  particular  event  will  always  be  the  cause  of  another,  we 
may  do  so  either  on  the  ground  that  it  is  the  known  cause  in 
this  instance,  or  that  its  causal  relation  is  the  result  of  a  real 
or  an  implied  inductive  inference.  In  the  former  case  the 
generalization  is  like  the  previous  one,  a  mere  enunciation  of 
what  is  implied  by  the  idea  of  cause.  In  the  latter  case  the 
generalization  is  a  probable  one,  but  might  be  considered  a 
deductive  inference  with  a  probable  conclusion,  on  the  ground 
that  the  supposition  of  the  particular  case  being  inductively 
an  instance  of  causality,  was  the  minor  premise  of  a  syllogism, 
in  which  the  universal  law  of  causation  was  the  major  premise. 
The  generalization  could  be  inductive  only  upon  the  assump- 
tion that  it  was  only  another  way  of  making  the  transition 
from  observed  facts  of  coincidence  or  sequence  to  necessary 
connection,  or  that,  instead  of  supposing  tacitly  or  openly  the 
existence  of  absolutely  the  same  conditions,  we  inferred  that  a 
particular  one  of  them  was  sufficient  in  the  future  to  produce 
the  effect.  This  last,  however,  is  really  based  on  the  assump- 
tion that  the  event  is  not  known  as  a  cause  in  the  first  place, 
but  that  we  are  still  dealing  with  it  merely  as  a  coincidence 
or  sequence. 

Let  us  take  an  illustration.     I  observe  in  one  or  more  in- 


326  ELEMENTS  OF  LOGIC 

stances  that  the  dew  falls  on  clear,  cool  nights.  If  I  infer 
that  clear,  cool  nights  will  always  cause  a  fall  of  dew,  I  evi- 
dently have  a  generalization  with  an  increment  of  conception 
not  involved  in  the  data  known,  and  one  which,  in  meaning, 
is,  that  all  clear,  cool  nights  will  produce  this  effect.  But 
whether  the  generalization  is  inductive  or  not  will  depend 
upon  the  question  whether  the  inference  turns  upon  the  uni- 
versality of  the  causal  nexus  in  such  cases,  or  the  probable 
similarity  of  all  other  (dear,  cool  nights  with  the  one  ob- 
served. If  it  turn  upon  the  universality  of  the  causal  nexus, 
this  may  be  merely  a  statement  of  what  is  implied  in  suppos- 
ing that  it  was  the  clearness  and  coolness  of  the  night  that 
caused  the  dewfall  in  the  observed  cases,  so  that  the  inductive 
inference  will  lie  in  the  mind's  implicit  or  explicit  transition 
fi-om  the  fact  of  coincidence  in  the  first  place  to  the  idea  of 
causal  connection,  and  not  in  universalizing  it.  On  the  other 
hand,  if  it  turn  upon  the  probable  similarity  in  other  respects 
of  all  other  clear  and  cool  nights  to  those  observed,  the  infer- 
ence may  be  inductive,  on  the  ground  that  it  is  not  from  the 
assumption  of  identity  in  all  the  conditions  to  the  effect,  but 
is  an  inference  to  the  single  efficiency  of  the  clearness  and 
coolness.  Of  course,  if  we  suppose  at  the  outset  that  the 
clearness  and  coolness  of  the  night  were  the  sole  causes  of  the 
dew  falling,  the  generalization  involving  all  such  nights  is 
only  an  explicit  enunciation  of  that  idea,  and  not  a  new  con- 
ception. But  usually  the  generalization  in  fact  means  the  se- 
lection of  those  qualities  out  of  all  that  are  present,  as  the 
sole  cause  of  the  phenomenon.  In  this  light  the  inference 
might  be  regarded  as  inductive  because  it  adds  to  the  ob- 
served possibility  of  a  number  of  conditions  the  conception 
that  a  particular  condition  is  the  sole  one  and  will  always  be 
the  only  one. 

These  illustrations  show  the  limitations  under  which  causal 
generalizations  are  to  be  regarded  as  inductive,  and  they  all 
practically  resolve  themselves  into  the  one  rule  that  dynamical 
inferences  are  inductive  only  when  the  transition  is  from  coin- 
cidences  or  sequences  to  causes.     The  mere  transition  from 


INDUCTIVE  REASONING  327 

individual  instances  to  the  universal  in  cases  involving  the 
causal  condition  is  not  necessarily  inductive,  because  it  is  com- 
plicated with  the  assumptions  about  reasoning  from  cause  to  ef- 
fect which  is  usually  spoken  of  by  logicians  as  deductive.  It 
is,  therefore,  necessary  in  such  cases  to  find  some  other  crite- 
rion than  the  mere  generalization  as  a  mark  of  what  the  nat- 
ure of  the  inference  is,  and  this  is  to  observe  whether  the  case 
involves  any  transition,  explicit  or  implicit,  from  merely  act- 
ual to  necessary  connection. 

The  possibility  of  a  confusion  of  deductive  with  inductive 
inferences  where  the  causal  nexus  is  involved,  is  brought  out 
admirably  by  Mr.  Venn  in  a  passage  of  his  "  Empirical  Logic," 
which  we  quote  in  illustration. 

"  A  man  is  bitten  by  a  cobra.  We  have  known  or  heard  of 
many  other  such  cases,  and  they  all  proved  fatal.  We  con- 
clude with  some  confidence  that  XY,  the  present  sufferer, 
will  die  ;  as  A,  B,  C,  the  former  ones,  are  all  supposed  to  have 
died.  Here  in  these  few  words  we  have  had  all  the  requisite 
facts  put  before  us,  and  we  also  have  the  inference  from  them. 

"  Now,  since  we  are  looking,  in  the  spirit  of  logicians,  at  the 
existence  of  this  belief,  which  we  know  will  inevitably  arise  in 
every  normal  mind,  we  proceed  to  exercise  what  Hume  calls 
'  our  sifting  humor,'  by  beginning  to  press  a  series  of  ques- 
tions. We  start  by  asking  the  observer  why  he  believes  in 
the  approaching  death  of  XY  ?  To  this  question  two  distinct 
answers  might  readily  be  given.  Some  would  say  off  hand, 
'  Because  every  one  who  is  so  bitten  always  dies  '  ;  others 
would  say,  'Because  A,  B,  C,  whom  we  know  to  have  been 
previously  bitten,  have  all  died.'  When  these  answers  are  ex- 
panded into  proper  shape  they  would  stand  respectively  as 
follows  : — 

"  Deductive. — All  men  who  are  bitten  die  :  the  man  XY  is 
bitten  :  therefore  XY  will  die. 

Ci  Inductive.— The  men  A,  B,  C,  were  bitten  and  died.  The 
man  XY  has  also  been  bitten.     Therefore  XY  will  die." 

This  illustration  is  only  to  say  that  the  inference  "XY  will 


328  ELEMENTS  OF  LOGIC 

die  "  may  be  either  inductive  or  deductive,  according  as  the 
premises  do  or  do  not  include  it.  In  the  second  form  it  is 
assumed  to  be  inductive  because  there  is  a  transition  from 
one  particular  case  or  set  of  cases  to  another,  so  that  the 
known  and  inferred  instance  may  ultimately  be  included  in  a 
wider  generalization  than  those  already  observed.  But  if  this 
individual  inference  be  a  deduction  from  the  tacit  assumption  that 
"all  persons  bitten  by  the  cobra  will  die,"  the  induction  origi- 
nally was  not  from  the  particular  known  cases  to  this  one,  but 
was  made  from  the  known  instances  to  this  universal  assumption 
by  supposing  the  connection  to  be  a  necessary  one  in  any  case. 
If  anyone  wishes  to  object  to  the  cogency  of  the  reasoning  in 
such  cases,  it  must  be  on  the  ground  that  the  assumption  made 
in  the  original  instance  was  not  proved,  or  that  the  inference 
is  only  an  inductive  one.  But  whether  it  is  inductive  or  de- 
ductive must  be  determined  by  first  deciding  what  are  the 
real  premises  in  the  case. 

This  is  sufficient  to  recommend  caution  in  regard  to  the  nat- 
ure of  inferences  in  particular  cases  respecting  the  causal  nexus 
and  generalizations  involving  this  idea.  The  present  treatise 
does  not  require  a  complete  exposition  of  the  conditions  and 
limitations  of  induction  in  this  respect.  The  student  must 
therefore  be  satisfied  with  the  ordinary  criterion,  that  causal 
generalizations  and  inferences,  to  be  inductive,  must  be  transi- 
tions from  what  is  conceived  as  actual  coincidences  to  neces- 
sary ones. 

The  Principles  of  Induction  are  the  next  subject  of  consid- 
eration. They  are  suggested  by  the  demand  to  know  why  or 
upon  what  ground  we  make  the  generalizations  just  discussed, 
or  inferences  extending  beyond  the  known  data  of  the 
premises. 

The  principle  supposed  to  lie  at  the  foundation  of  inductive 
reasoning  is  sometimes  announced  as  the  Law  of  Universal 
Causation,  and  sometimes  the  Law  of  the  Uniformity  of  Nature. 
There  is  an  important  difference  between  these  two  principles 
which  we  shall  have  to  notice,  but  it  is  not  such  as  to  affect 
seriously  the  theory  of  induction.     The  latter  form  of  state- 


INDUCTIVE  REASONING  329 

ment  is  adopted  by  those  who  feel  doubtful  about  the  exist- 
ence of  any  such  necessary  connection  as  is  implied  by  the  no- 
tion of  "  cause."  The  difference  between  the  two  conceptions 
can  be  brought  out  by  their  definition. 

The  Law  of  Universal  Causation  means  that  every  event  must 
have  a  cause,  and  every  cause  must  have  an  effect.  This  im- 
plies that  the  mind  is  not  satisfied  with  the  mere  existence  or 
occurrence  of  a  phenomenon,  but  must  know  upon  what  it  de- 
pends, the  ground  for  its  being  what  it  is.  The  principle  is 
usually  spoken  of  as  an  d  priori  one,  which  means  that  we  must 
assume  it  in  all  our  thinking  and  can  give  no  proof  of  it.  As 
Logic  is  not  concerned  with  proving  that  it  is  a  priori,  or  with 
the  speculations  centering  about  the  use  of  the  term,  we  do 
not  enter  into  these  discussions,  but  must  content  ourselves 
with  recognizing  that  Logic  and  logical  processes  take  the 
law  for  granted. 

The  Law  of  the  Uniformity  of  Nature  means  that  the  phe- 
nomena of  the  world  present  a  certain  uniformity  of  qualities  or 
occurrence  that  enables  us  to  conceive  it  as  a  system  of  them  hav- 
ing a  definite  unity  and  order.  The  law  differs  from  that  of 
causation  in  not  representing  a  necessary  assumption  or  prin- 
ciple of  the  mind,  but  in  expressing  the  observed  facts  of  ex- 
perience. "We  must  think  of  events  as  having  a  cause,  whether 
they  occur  regularly  or  not.  But  the  discovery  of  their  ac- 
tual uniformity,  whether  of  resemblance  or  occurrence,  is  a 
matter  of  actual  observation  and  experience.  The  law  is 
therefore  called  empirical,  in  contrast  with  the  term  d  priori 
describing  the  former  law. 

In  regard  to  their  relation  to  Induction,  the  two  laws  may 
be  said  to  be  complementary  of  each  other,  as  we  shall  ex- 
plain. But  it  is  important  to  remark  that  they  are  not  ex- 
clusively principles  regulating  the  process  of  induction.  They 
are  both  equally  related  to  deductive  reasoning.  Thus  the 
first  law,  the  Law  of  Universal  Causation,  is  only  another 
statement  of  the  Law  of  Sufficient  Reason,  which  we  have  dis- 
cussed in  the  chapter  on  the  Laws  of  Thought.  It  is  true 
that  some  writers  would  limit  the  Law  of  Sufficient  Reason  to 


330  ELEMENTS  OF  LOGIC 

inductive  processes,  but  the  present  writer  sees  no  ground  for 
excluding  the  same  law  from  a  determining  influence  on  thte 
deductive  inference.  The  main  distinction  is  that  the  law  is 
applied  in  a  somewhat  different  way  in  the  two  processes. 
But  it  is  not  important  in  this  treatise  to  discuss  the  question, 
nor  even  to  take  any  positive  attitude  upon  one  side  or  the 
other  of  it.  It  is  sufficient  to  know  the  general  ojnnion  of 
logicians,  that  the  law  of  causation  is  an  accepted  principle  af- 
fecting inductive  inferences. 

The  manner  in  which  the  law  affects  induction  is  this  : 
There  is  no  reason  in  the  fact  itself  why  we  should  go  beyond 
the  data  of  the  premises.  "When  we  infer  something  which 
has  not  been  stated  in  the  premises  we  must  know  why  we  are 
so  disposed  to  act,  or  upon  what  ground  we  thus  go  from  the 
known  to  the  unknown,  from  the  particular  to  the  universal, 
etc.  The  only  ground  or  reason  assignable  is  that  there  is,  in 
the  circumstances  involved,  a  phenomenon  requiring  explana- 
tion, and  that  the  most  probable  one  in  the  case  is  to  be 
found  in  the  inference  drawn.  Thus  I  observe  a  very  marked 
resemblance  between  certain  phenomena  of  sound  and  light, 
and  I  know  that  in  the  case  of  sound  this  phenomenon  is  con- 
nected with  its  nature  as  vibrations.  This  resemblance  re- 
quires explanation  because  every  event  must  be  supposed  to 
have  its  cause.  In  this  special  instance  the  resemblance  of 
the  phenomenon,  and  its  known  connections  in  the  other,  are 
taken  as  facts  justifying  the  inference  that  undulations  in  light 
will  account  for  a  phenomenon  which  is  essentially  the  same 
as  that  which  is  accounted  for  in  the  same  way  when  occur- 
ring in  the  case  of  sound. 

But  the  mere  rationality  of  the  demand  for  an  explanation 
of  a  resemblance,  a  difference,  a  coincidence,  or  a  sequence,  is 
not  a  proof  that  the  inference  or  conjecture  is  the  right  one. 
It  only  explains  why  the  mind  seeks  to  go  beyond  the  known 
facts,  and  guarantees  only  that  the  phenomena  shall  have  some 
cause,  but  does  not  indicate  even  with  a  degree  of  probability 
what  particular  cause  it  shall  be.  Another  law  must  come  in 
at  this  point  to  determine  what  inference  shall  be  the  probable 


INDUCTIVE  REASONING  331 

one.  This  is  the  Law  of  the  Uniformity  of  Nature,  or  the  act- 
ually observed  frequency  with  which  certain  phenomena  have 
been  connected  in  our  experience.  This  law  decides  the 
greater  probability  of  any  given  inference  both  by  itself  and  in 
relation  to  other  possible  inferences  at  the  same  time.  The 
occurrence  of  a  resemblance  between  two  bodies,  say  of  elec- 
trical effects  in  the  case  of  electricity  and  magnetism,  on  a 
single  occasion,  might  justify  under  the  law  of  causation  the 
supposition  of  their  possible  identity.  But  owing  to  an  equal 
possibility  that  the  resemblance  was  onhy  accidental,  the  infer- 
ence would  be  a  weak  one,  unless  we  were  able  to  give  it  some 
probability  from  the  peculiar  nature  of  the  resemblance  ;  and 
this  characteristic  is  often  a  factor  in  such  cases,  but  has  no 
rules  for  determining  when  it  does  and  when  it  does  not  give 
probability.  But  if  we  frequently,  and  under  all  sorts  of  vary- 
ing circumstances,  have  observed  this  resemblance  between  the 
two  phenomena,  the  probability  of  their  identity  is  greatly  in- 
creased. The  law  of  causation  determines  the  possibility  of 
the  identity,  and  even  the  likeliness  that  the  same  effect  has 
the  same  cause ;  but  the  law  of  the  actual  uniformity  of  nature 
determines  the  probability  that  the  cause  is  to  be  found  in 
these  particular  bodies.  The  frequency  with  which  peculiar 
resemblances  and  coincidences,  differences  and  sequences  oc- 
cur affords  a  probability  not  only  of  their  reoccurrence  in  the 
same  connection,  but  also  of  the  fact  that  this  cause  will  be 
found  in  some  of  the  known  causal  agencies  in  connection  with 
the  objects  manifesting  the  phenomena. 

But  in  addition  to  the  two  general  principles  thus  enunci- 
ated, there  are  two  subordinate  principles  or  canons  which  are 
determinative  of  the  process,  and  which  are,  in  a  measure  at 
least,  corollaries  of  the  general  law  of  the  uniformity  of  nature, 
and  are  mainly  the  means  of  determining  the  legitimacy  of 
the  inductive  inference.  I  shall  call  them  the  Principle  or 
Canon  of  Agreement,  and  the  Principle  or  Canon  of  Differ- 
ence. These  are  practically  the  same  as  the  Methods  of 
Agreement  and  Difference  so  denominated  by  logicians  gener- 
ally, only  I  wish  here  to  distinguish  them  as  organs  of  discov- 


332  ELEMENTS  OF  LOGIC 

ery,  from  their  frequent  and  actual  application  in  connection 
with  deductive  assumptions,  as  verifications  of  inductive  infer- 
ences already  made.  This  latter  use  of  them  we  shall  com- 
ment upon  again. 

The  Principle  of  Agreement  can  be  briefly  defined  as  the 
principle  which  determines  the  probability  of  a  given  identity 
or  connection  on  the  ground  of  the  actual  frequency  of  certain 
resemblances  or  coincidences  under  varying  conditions.  Or 
more  simply  still,  the  agreement  of  two  phenomena  in  respect 
of  the  qualities  producing  them,  or  in  respect  of  the  connection 
in  which  they  occur  is  a  criterion  of  their  cause.  Thus,  to  il- 
lustrate, if  A,  B,  C,  and  D,  E,  F,  resemble  each  other  in  a  cer- 
tain marked  phenomenon  or  quality  known  to  be  characteristic 
of  each  object,  the  probability  that  they  are  identical  in  their 
nature,  structure,  functions,  etc.,  is  proportioned  to  the  de- 
gree of  resemblance,  the  frequency  of  the  phenomenon's  occur- 
rence, and  the  assurance  we  feel  about  the  relation  between 
particular  causes  and  effects.  If  A,  B,  C,  and  D,  E,  F,  repre- 
sent events  associated  together,  the  probability  that  they  are 
necessarily  connected  is  proportioned  to  the  frequency  with 
which  they  occur  together  under  varying  conditions.  "  For 
example,  bright  prismatic  colors  are  seen  on  bubbles,  on  films 
of  tar  floating  upon  water,  on  thin  plates  of  mica,  as  also  on 
cracks  in  glass,  or  between  two  pieces  of  glass  pressed  to- 
gether. On  examining  all  such  cases  they  seem  to  agree  in 
nothing  but  the  presence  of  a  very  thin  layer  or  plate,  and  it 
appears  to  make  no  appreciable  difference  of  what  kind  of 
matter,  solid,  liquid,  or  gaseous,  the  plate  is  made.  Hence  we 
conclude  that  such  colors  are  caused  merely  by  the  thinness 
of  the  plates,  and  this  conclusion  is  proved  true  by  the  theory 
of  the  interference  of  light.  Sir  David  Brewster  beautifully 
proved  in  a  similar  way  that  the  colors  seen  upon  mother-of- 
pearl  are  not  caused  by  the  nature  of  the  substance,  but  by  the 
form  of  the  surface.  He  took  impressions  of  the  mother-of- 
pearl  in  wax,  and  found  that  although  the  substance  was  en- 
tirely different  the  colors  were  exactly  the  same.  And  it  was 
afterward  found  that  if  a  plate  of  metal  had  a  surface  marked 


INDUCTIVE  REASONING  333 

by  very  fine  grooves,  it  would  have  iridescent  colors  like  those 
of  mother-of-pearl." 

It  should  be  remarked,  respecting  these  examples,  that  they 
also  contain  illustrations  of  the  Canon  of  Difference,  as  all 
cases  of  agreement  actually  do  which  have  any  conclusiveness 
at  all.  But  the  important  thing  to  be  observed  in  them  is  that 
the  inference  is  first  suggested  by  a  number  of  resemblances 
where  they  might  not  be  suspected  ;  that  is,  in  conjunction 
with  differences  that  make  the  resemblances  significant. 

The  Canon  of  Difference  lays  the  emphasis  upon  the  differ- 
ences between  groups  of  phenomena,  and  so  infers  a  causal 
connection  from  the  separation  of  certain  phenomena  from 
others.  It  means  that,  if  two  phenomena  are  constantly  iso- 
lated together  from  other  groups  which  remain  invariable,  the 
separated  phenomena  may  be  taken  as  necessarily  connected 
in  the  relation  of  cause  and  effect ;  that  which  is  known  to  be 
the  antecedent  being  the  cause,  and  the  consequent  the  effect. 
An  example  of  this  method  is  given  in  the  case  quoted  from 
Mr.  Fowler,  in  which  the  isolated  action  of  gravity  is  inferred 
to  explain  the  equal  velocities  of  bodies  falling  through 
equal  spaces  in  equal  times.  Ordinarily,  bodies  of  different 
weights,  where  this  difference  is  considerable — say  lead  and 
feathers — show  marked  differences  of  velocity  in  falling,  and  it 
was  inferred  that  gravity  did  not  affect  all  bodies  equally. 
Here  was  a  case  of  differences,  however,  which  did  not  take 
into  account  the  uniformly  accompanying  fact  of  resistance 
from  the  air.  But  the  separation  of  this  influence  and  the  con- 
comitant isolation  of  gravity  and  of  the  falling  bodies  gives 
rise  to  another  inference,  and  the  very  opposite  of  the  previ- 
ous one,  upon  the  basis  of  the  difference  between  this  phe- 
nomenon and  others.  The  inference  is  conclusive  in  projior- 
tion  to  the  certitude  that  the  conditions  are  as  we  suppose 
them.  If  we  had  not  felt  in  isolating  gravity  that  there  could 
be  no  other  disturbing  factor  at  the  time,  our  inference  would 
have  been  liable  to  the  same  error  as  that  which  had  been 
based  upon  the  actual  differences  of  velocity  in  falling  bodies. 

The   Principle  of  Difference  is  complementary  to   that   of 


334  ELEMENTS  OF  LOGIC 

Agreement,  and  is  often,  if  not  always,  found  in  connection 
with  it,  except  that  the  inference  does  not  turn  upon  the 
agreements,  but  upon  the  differences.  But  the  differences  are 
noticeable  directly  in  proportion  to  the  invariability  of  other 
phenomena  without  those  differences,  and  hence  such  circum- 
stances are  a  great  indirect  help  to  the  inference.  In  the 
example  quoted  to  illustrate  the  Principle  of  Agreement,  it 
was  certain  differences  between  the  instances  that  made  the 
inference  drawn  a  possible  one.  The  differences  between  the 
natures  of  the  various  substances  while  the  iridescence  was  a 
common  quality  showed  very  distinctly  that  the  peculiar  kind 
of  substance  had  nothing  to  do  with  the  effect,  and  this  nega- 
tive inference  had  as  important  a  part  in  determining  the  in- 
ference as  the  agreement  of  the  various  substances  in  reflecting 
light.  So,  in  the  example  quoted  from  Mr.  Fowler,  the  agree- 
ment between  the  retarding  influence  of  the  air  and  the  de- 
creased velocity  of  the  lighter  body,  and  the  uniformity  of 
opposite  effects  without  the  isolation  of  gravity,  are  as  neces- 
sary to  the  inductive  inference  as  the  variation  from  the  usual 
order  when  gravity  is  isolated.  But  the  explicit  ground  of 
the  inference  depends  in  the  one  instance  upon  the  agreement, 
and  in  the  other  upon  the  difference,  between  the  phenomena. 
It  must  be  remarked  regarding  the  Principle  of  Difference, 
that  it  is  most  frequently  applicable  in  cases  of  experiment, 
and  experiment  belongs  more  properly  to  verification  rather 
than  discovery,  or  to  confirming  the  inference  rather  than  first 
suggesting  it.  This  is  not  exclusively  the  case,  however.  The 
same  remark  is  not  so  true  of  the  Principle  of  Agreement, 
which  is  more  frequently  the  incident  of  discovery,  although 
it  can  be  made,  and  often  is,  the  instrument  of  verification  and 
experiment.  It  is  when  the  two  principles  are  combined  with 
observation  and  experiment  artificially  applied,  and  with  de- 
ductive principles  assumed  or  proved,  that  they  become  means 
of  verification  rather  than  the  means  of  originating  the  induc- 
tive inference  ;  and  it  is  this  fact  which  makes  it  so  difficult,  if 
not  impossible,  to  distinguish  as  accurately  as  is  desirable  be- 
tween pure  inductive  reasoning  and  scientific  method  or  verifi- 


INDUCTIVE  REASONING  335 

cation.  They  are  regulative  of  inductive  inferences  pure  and 
simple,  when  the  reasoning  is  suggested  by  them  alone,  and  not 
complicated  with  assumed  or  known  facts  or  principles  in- 
volving an  application  of  the  Laws  of  Identity  and  Contradic- 
tion. As  principles,  however,  they  are  only  modifications  of 
the  Uniformity  of  Nature.* 

*  For  general  references  on  the  nature  of  induction,  the  following 
works  may  be  consulted  :  Mills:  Logic,  Bk.  III.,  Chaps.  I.  to  V.  inclu- 
sive. Hamilton  :  Lectures  on  Logic,  Lects.  XVII.  and  XXXIII.  Venn  : 
Empirical  Logic,  Chaps.  XIV.  and  XV.  Jevons :  Principles  of  Science, 
Bk.  I.,  Chap.  VII.  ;  Bk.  II.,  Chap.  XL  (the  whole  book  should  be  care- 
fully read).  Fowler  ;  Inductive  Logic,  Chap.  I.  Ueberweg  :  System  of 
Logic  and  History  of  Logical  Doctrine,  Sections  127,  128,  and  129,  pp. 
476-490. 


CHAPTER  XXin. 

SCIENTIFIC  METHOD 

I  THE  NATURE  OF  SCIENTIFIC  METHOD.— We  have 
hinted  at  the  distinction  between  inductive  reasoning  and 
scientific  method,  and  perhaps  we  have  implied  that  the  latter 
is  limited  to  the  verification  of  the  former.  So  far  as  scien- 
tific method  is  identified  only  with  what  is  usually  called  "In- 
ductive Method,"  this  is  perhaps  the  case.  But  there  is  a 
broader  use  of  the  conception,  which  involves  also  the  applica- 
tion of  deductive  principles,  which,  besides  being  distin- 
guished from  purely  inductive  reasoning,  may  appear  in  com- 
bination with  the  inductive,  or  be  a  deductive  process  solely. 
This  broader  use  of  the  conception  gives  rise  to  two  subordi- 
nate forms  of  logical  procedure.     They  are  usually  called  : 

(1)  The  Method  of  Discovery. 

(2)  The  Method  of  Instruction. 

The  method  of  discovery  is  occupied  with  the  acquisition  of 
knowledge,  and  is  usually  identified  with  Induction  or  the  In- 
ductive Method,  as  it  is  generally  called.  The  method  of  in- 
struction is  occupied  with  the  communication  of  knowledge. 
But  the  name  purports  to  include  more  than  the  mere  im- 
parting of  truth  once  discovered,  and  hence  it  is  meant  to  be 
identified  with  what  is  called  the  Deductive  Method,  which 
aims  to  prove  as  well  as  to  impart  truth.  But  the  two 
methods  are  not  wholly  independent  of  each  other,  as  the  se- 
quel of  the  present  discussion  is  intended  to  show.  They 
may  be  combined  in  certain  stages,  both  of  discovery  and  in- 
struction. This  requires  us  to  recognize  the  divisions  only 
for  provisional  purposes,  and  to  indicate  processes  which  do 
not  exactly  coincide.     In  certain  features  the  two  methods  are 


SCIENTIFIC  METHOD  337 

entirely  distinct  from  each  other.  It  is  when  they  are  both 
implicated  in  the  discovery  and  establishment  of  the  same 
truth,  that  they  together  form  the  perfect  application  of  scien- 
tific method.  We  shall  call  them  respectively  the  Inductive 
and  the  Deductive  Methods,  and  examine  the  deductive  first 
in  order. 

//.  THE  DEDUCTIVE  METHOD.— The  deductive  method 
has  to  do  with  the  communication  and  proof  of  existing 
knowledge.  The  knowledge  may  exist  as  a  positive  acquisi- 
tion, or  be  positively  known,  only  by  the  person  imparting  it,  or 
proving  it,  and  so  involve  something  of  discovery  to  the  per- 
son receiving  it.  But  this  fact  does  not  prevent  the  mental 
processes  of  the  receiver  from  being  deductive  as  well  as 
those  of  the  importer.  The  object  of  the  method  is  to  assure 
the  truth  of  the  matter  concerned,  to  give  it  more  than  a 
probable  value.  The  method  comprehends  three  distinct  pro- 
cesses :  Definition,  Division,  and  Probation. 

1st.  Definition. — Definition  is  the  process  of  making  clear 
all  the  conceptions  entering  into  the  thesis  or  proposition  to 
be  proved.  It  unfolds  their  intension.  The  first  thing,  of 
course,  is  to  know  what  is  to  be  established,  and  the  method 
of  proving  it  begins  with  the  definition  of  the  elements  of  the 
judgment  or  thesis  enunciated  for  proof.  The  nature  and 
principles  of  definition  have  already  been  discussed,  and  need 
not  be  repeated  here.  The  process  itself  does  for  the  concep- 
tions of  a  proposition  what  proof  does  for  the  proposition  it- 
self. 

2d.  Division. — Division  is  the  process  of  rendering  our 
conceptions  more  definite  in  respect  to  their  relations  to  each 
other.  It  is  complementary  to  Definition,  and  so  unfolds  the 
extension  of  a  notion.  It  is  to  some  extent  implied  in  Defini- 
tion, but  can  go  beyond  that  by  showing  the  exhaustive  na- 
ture of  our  ideas,  and  the  amount  of  truth  involved  in  the 
thesis  to  be  proved.  This  subject  has  also  been  duly  treated 
in  its  proper  place,  and  requires  no  further  mention. 

To  illustrate  what  is  meant  by  Definition  and  Division,  as 
aids  to  instruction  and  proof,  suppose  it  is  required  to  estab- 
22 


338  ELEMENTS  OF  LOGIC 

lish  the  truth  of  the  proposition,  "  Governments  are  useful." 
I  must  first  define  what  "Government  "  is,  and  what  "utility" 
is.  In  so  doing  I  indicate  the  characteristics  constituting 
them,  conferential  and  differential.  But  this  does  not  tell  me 
how  much  is  involved  in  the  proof  of  the  thesis.  I  must  di- 
vide the  two  terms  into  their  species  ;  "  government,"  for  in- 
stance, into  monarchic,  oligarchic,  and  democratic,  or  other 
such  species  as  the  convenience  of  the  argument  may  require. 
This  will  show  more  distinctly  what  is  involved  in  proving  the 
main  thesis.  So  also  if  "  utility  "  be  divided  into  species, 
such  as  economic,  artistic,  scientific,  etc.  The  two  processes, 
definition  and  division,  bring  out  more  clearly  than  one  of 
them  alone,  the  conferentia  and  differentia,  genus  and  species, 
or  qualitative  and  quantitative  aspects  of  our  conceptions." 

3d.  Probation. — Probation  is  the  process  of  proof;  or  the 
statement  of  certain  truths  which  render  the  thesis  a  neces- 
sary conclusion  from  them.  The  thesis  is  the  proposition  to 
be  proved.  The  truths  which  prove  it  are  the  known  facts  and 
principles  which  may  constitute  the  premises,  and  the  thesis 
will  be  the  conclusion.  These  determining  truths  may  be 
axioms,  postulates,  proved  propositions,  or  any  truth  which 
the  person  to  whom  the  probation  is  made  may  accept.  Their 
acceptance  is  the  condition  of  their  proving  anything.  We 
must  observe,  therefore,  that  probation  is  a  material  as  well  as 
a  formal  process.  It  is  always  syllogistic,  but,  as  Hamilton 
observes,  the  converse  is  not  necessarily  true  ;  namely,  that 
all  syllogistic  reasoning  is  probation.  The  object  in  probation 
is  to  prove  the  material  truth  of  the  thesis,  and  not  to  con- 
duct merely  a  formal  process  of  reasoning.  We  must  there- 
fore enunciate  some  facts  or  principles  accepted  by  the  per- 
son to  whom  the  probation  is  made,  and  then  bring  the 
thesis  under  it  in  such  a  way  that  it  will  be  a  necessary  con- 
sequence of  what  is  already  admitted.  In  this  way  we 
give  absolute  certainty  to  our  proposition,  provided  we  avoid 
the  usual  fallacies  in  syllogistic  reasoning. 

To  illustrate,  suppose  I  am  required  to  prove  that  the  sum 
of  the  angles  in  a  triangle  is  equal  to  two  right  angles.     I 


SCIENTIFIC  METHOD  339 

must  first  define  the  terms  involved,  namely,  "angle,"  "  right 
angle,"  "  triangle,"  etc.,  and  then  I  must  announce  the  funda- 
mental principles  that  are  either  assumed  or  admitted  in  the 
process  of  proof.  There  may  be  such  axioms  and  postulates 
as  that  things  equal  to  the  same  thing  are  equal  to  each 
other,  all  right  angles  are  equal  to  each  other,  and  all  tri- 
angles are  essentially  identical  in  their  properties,  etc.  If  then 
I  can  show,  either  by  observation  or  proof,  that  the  sum  of  the 
angles  of  the  triangle  is  equal  to  some  quantity  which  is  known 
or  admitted  immediately  to  be  equal  to  two  right  angles,  the 
major  premise  involved  in  the  axiom  mentioned  insures  the 
conclusion  I  am  required  to  draw.  The  proposition  may  not 
carry  with  it  its  own  evidence,  but  the  truth  of  some  prior 
proposition  or  propositions  may  prove  it.  The  following  dia- 
gram and  argument  illustrates  the  whole  method.     The  the- 


Fm.  31. 

sis  to  be  proved  is  that  a  +  b  +  c  =  2  right  angles.  It  is  as- 
sumed that  c  and  d  +  e  are  right  angles  and  equal  to  each 
other.  Hence  c  +  d  +  e  —  2  right  angles.  By  construction 
or  previous  proof,  b  is  taken  to  be  equal  to  d  and  a  to  e,  so 
that  a  +  b  =  d  +  e,  a  right  angle  ;  a  +  b  +  c,  therefore,  equals 
c  +  d  +  e,  and  by  assumption  c  +  d  +  e  —  2  right  angles. 
Hence  whoever  admits  the  previous  steps  and  conditions  must 
admit  that  a  +  b  +  c  =  2  right  angles. 

The  process  would  be  quite  similar  to  this  in  any  other 
proposition,  such  as  "  civil  law  is  necessary  to  the  preserva- 
tion of  order."  If  this  statement  did  not  evince  its  own 
truth,  we  should  be  obliged  to  announce  some  acceptable 
truth  which  carried  the  given  conclusion  with  it.  Thus  we 
might  show  that  restrictions  of  individual  liberty  were  neces- 
sary to  this  end,  and  this  assertion  might  be  admitted  with- 
out proof,  but  on  its  own  transparency.  Then,  if  the  defini- 
tion of  civil  law  showed  it  to  be  the  only  restriction  adapted  to 


340  ELEMENTS  OF  LOGIC 

such  an  end,  the  truth  of  our  proposition  must  follow  as  an 
inference,  whether  the  person  to  whom  the  proof  was  directed 
could  fully  see  the  contents  and  meaning  of  the  conclusion  or 
not. 

It  is  not  always  essential  that  the  deductive  method  should 
take  an  explicit  syllogistic  form,  because  one  of  the  premises 
may  be  so  apparent  that  the  conclusion  would  follow  upon  the 
enunciation  of  a  single  conditioning  truth,  or  it  may  follow  as 
an  immediate  inference  from  the  conditioning  proposition. 
Usually,  however,  a  syllogism  is  implied.  In  disproof,  which 
involves  the  same  general  method,  we  state  or  assume  the  con- 
tradictory of  the  thesis,  as  in  probation  we  state  the  condition- 
ing proposition.  These  serve  to  measure  the  truth  of  a  given 
assertion  by  its  agreement  or  its  contradiction  with  truths  al- 
ready known  or  accepted.  No  guessing,  or  probabilities,  based 
upon  the  actual  uniformities  of  nature,  enter  into  the  case,  ex- 
cept as  they  might  have  originally  determined  the  proposition 
which  is  assumed  as  proof.  The  proof  depends  upon  the  ac- 
ceptance of  the  principles,  and  not  upon  the  method  by  which 
they  were  originally  suggested.  We  take  up  the  inductive 
method  next. 

III.  THE  INDUCTIVE  METTIOD.-Mtev  what  has  been  said 
about  the  meaning  of  the  term  "  induction,"  it  might  seem  ob- 
jectionable to  use  the  term  for  describing  a  method  which  is 
to  be  sharply  distinguished  in  some  of  its  features  from  the 
process  of  reasoning  going  by  that  name.  But  as  long  as  we 
are  careful  to  distinguish  between  induction  or  inductive  rea- 
soniug  as  a  mode  of  inference,  and  the  process  usually  called 
"  the  inductive  method  "  and  comprehending  the  principles  of 
verification,  there  will  be  no  serious  reason  for  rejecting  the 
traditional  vise  of  the  term  ;  especially  as  the  method  is  mainly, 
if  not  wholly,  occupied  with  the  acquisition  of  knowledge.  It 
is  often  identified  with  scientific  method,  which,  as  we  have 
explained,  is  really  a  combination  of  inductive  and  deductive 
processes  in  many  of  the  cases  assumed  to  be  purely  deductive. 
Because  of  this  fact  and  general  usage,  we  shall  treat,  in  this 
section,  of  "  the  inductive  method  "  as  usually  conceived,  and 


SCIENTIFIC  METHOD  341 

without  trying  to  push  the  distinction  between  inductive  rea- 
soning and  the  complications  of  induction  and  deduction  in  the 
process  of  verification  into  any  new  departures  of  phraseology. 
Were  we  discussing  the  subject  for  any  other  than  elementary 
purposes,  we  might  be  tempted  by  innovations.  But  they  are 
not  necessary  here,  and  hence  we  shall  employ  the  expression  to 
include  all  the  steps  involved  in  the  passage  from  facts  to  general 
ideas  and  their  verification.  The  process  so  named  includes  an 
exposition  of  the  preliminary  methods  or  inductive  inferences, 
the  four  inductive  methods  and  the  verification  therein  in- 
volved, along  with  the  frequent  application  of  deductive  princi- 
j)les  which  give  greater  assurance  to  truths  first  suggested  by 
induction.  In  other  words,  we  use  the  expression  here  to  in- 
dicate all  the  processes  involved  in  the  acquisition  and  verifica- 
tion of  knowledge  that  is  not  deduced  solely  from  previous 
conceptions.  These  processes  may  be  reduced  to  two  general 
classes,  somewhat  merging  into  each  other  in  actual  experi- 
ence. They  may  be  called  Acquisition  and  Verification,  pri- 
mary and  secondary  conditions,  or  subsidiary  and  confirmatory 
conditions  of  new  inferential  knowledge.  Their  import  and 
the  extent  of  their  application  will  appear  on  examination.  It 
must  be  observed  that  we  are  here  dealing  only  with  new  in- 
ferential truth,  and  not  with  what  may  be  acquired  by  non- 
logical  processes.  In  scientific  method,  however,  we  shall  be 
permitted  to  mention  the  conditions  or  data  upon  which  such 
inferences  are  based.  Some  of  them  are  included  under  the 
subsidiary  processes  to  induction,  as  they  are  called  by  writers 
generally. 

1st.  Acquisition,  or  Primary  Stages  of  Ascertaining  Mew 
Facts  and  Truths. — These  are  so  called  because  they  are  the 
first  conditions  of  the  truths  which  turn  out  to  be  acquired  by 
a  method  which  is  not  deductive,  or  which  does  not  represent 
in  the  first  stages  of  its  conception  any  derivation  from  estab- 
lished  knowledge.  In  the  acquisition  of  knowledge  in  general, 
there  are  of  course  more  processes  involved  than  mere  infer- 
ence or  reasoning.  Indeed,  the  first  and  most  important 
means  of   attaining  such  are  entirely  prior  to   any  form   of 


342  ELKMKXTS   OF   LOCK1 

reasoning  whatever.  They  are  the  common  processes  of  Per- 
ception and  Introspection,  by  which  we  simply  observe  facts 
of  experience,  and  perhaps  group  them  together  into  cla 
by  comparison  and  generalization.  But  these  are  processes 
whose  analysis  and  investigation  belong  more  properly  to  Psy- 
chology. Yet  it  is  not  possible  to  understand  their  relation  to 
the  properly  logical  processes  of  the  mind  unless  we  mention 
them,  and  indicate  the  manner  in  which  they  <  it  her  condition 
the  inductive  inference,  or  confirm  it  when  made.  We  there- 
fore begin  with  a  brief  consideration  of  these  processes  as  fur- 
nishing the  data  of  scientific  method. 

1.  Observation. — Observation  is  simply  the  perception  of 
facts.  It  denotes  to  watch  for,  and  merely  involves  what  is 
contained  in  perception  and  introspection.  The  data  from 
which  to  make  inferences  must  first  be  known,  and  observation 
is  the  means  of  ascertaining  them.  The  amount  of  our  knowl- 
edge is  determined  by  its  accuracy.  Those  who  are  good  ob- 
servers will  discover  more  data  for  inductive  inferences  than 
those  who  are  not.  Observation  may  be  of  two  kinds,  spon- 
taneous or  casual,  and  voluntary  or  rational.  The  former  re- 
quires no  distinct  effort  on  the  part  of  the  person  observing  ; 
the  latter  requires  an  act  of  attention  and  is  frequently,  if  not 
always,  employed  to  verify  some  expected  or  anticipated  phe- 
nomenon. The  progress  of  science  depends  much  more  upon 
rational  than  upon  casual  observation. 

Experiment  is  generally  mentioned  as  a  subsidiary  process 
to  induction.  This  is  often  the  case  in  regard  to  particular 
kinds  of  induction,  but  it  is  not  a  condition  of  all  inductions, 
nor  of  the  process  in  its  earliest  forms  It  is,  however,  a  very 
important  means  to  increasing  the  data  for  inductive  infer- 
ences. It  is  not  an  instrument  which  can  be  employed  inde- 
pendently of  observation,  but  is  only  a  help  to  that  process. 
It  is  the  means  both  of  varying  and  multiplying  observations. 
It  is  the  artificial  production  or  repetition  of  natural  phe- 
nomena for  the  purpose  of  insuring  the  accuracy  of  first  ob- 
servations or  of  correcting  them.  It  is  therefore  one  of  the 
incidents  of  observation  in  its  highest  forms.     "  Mr.  Mill  dis- 


SCIENTIFIC  METHOD  343 

tinguishes  between  the  two  processes  by  saying  that  in  obser- 
vation we  find  our  instance  in  nature  ;  in  experiments  we 
make  it  by  an  artificial  arrangement  of  circumstances.  When, 
as  in  astronomy,  we  endeavor  to  ascertain  causes  by  simply 
watching  their  effects,  we  observe  ;  when,  as  in  our  laboratories, 
we  interfere  arbitrarily  with  the  causes  or  circumstances  of  a 
phenomenon,  we  are  said  to  experiment."''  Experiment  may 
also  be  used  as  a  means  of  verification  as  well  as  acquisition, 
and  so  also  can  observation.  But  of  these  functions  we  shall 
speak  in  the  proper  place.  It  is  sufficient  to  remark  of  the  two 
that  they  occupy,  as  subsidiary  processes,  the  position  in  in- 
ductive method  that  is  held  by  definition  in  deductive  method. 

There  are  two  rides  for  regulating  the  use  of  observation  as 
an  instrument  of  scientific  discovery,  in  so  far  as  it  is  merely 
a  subsidiary  process  to  inductive  inferences.     They  are  : 

(a)  The  observations  should  be  precise. 

(6)  They  should  be  relevant  or  material. 
The  first  rule  enjoins  accuracy,  and  the  second  the  disposition 
to  seek  and  see  what  is  essential  and  important. 

2.  Classification. — This  process  does  the  same  for  induc- 
tive methods  that  Division  does  for  the  deductive,  and  it  is 
also  the  reverse  of  Division.  It  is  the  process  of  reducing 
phenomena  to  systematic  groups  on  the  ground  of  their  re- 
semblances and  invariable  connections.  It  is  the  means  of 
keeping  together  all  relevant  and  material  circumstances  in 
connection  with  objects  and  phenomena,  and  hence  is  of  great 
help  in  determining  tbe  extent  of  the  inferences  we  draw  on 
particular  occasions,  as  well  as  assigning  the  limits  within 
which  they  can  be  drawn.  It  is  the  reduction  of  phenomena  to 
their  classes  or  genera,  as  Division  reduces  genera  to  species. 
But  the  process  need  not  be  dwelt  upon  at  length  in  this 
treatise.     We  turn  to  the  third  and  more  important  process. 

3.  Hypothesis,  or  the  Forming  of  Hypotheses. — An  hypo- 
thesis may  be  briefly  defined  as  a  supposition.  It  is  an  infer- 
ence from  certain  facts  to  others,  or  to  the  cause  of  those 
which  are  matters  of  observation.  This  account  of  it  identi- 
fies it  with  the  inductive  inference,  and  this  is  our  intention. 


344  ELBMEJS  TB  OF  LOGIC 

Some  would  distinguish  it  from  "  induction. "  But  the  dis- 
tinction, when  examined,  only  turns  out  to  be  one  between  a 
supposition  expressing  ;i  mere  possibility  and  one  expressing 

a  probability  of  such  a  degree  as  to  set  aside  the  legitimacy  of 
other  suppositions,  at  leasl  for  the  time.  An  hypothesis  simply 
supposes,  in  it.^  primary  stage,  a  possible  facl  or  cause  I 
plain  phenomena,  and  may  turn  out  to  be  tenable  or  uot,  ac- 
cording as  Further  observation  may  determine.  But  an  "in- 
duction "  is  supposed  to  represent  a  firmer  mental  allegiance 
limn  hypothesis,  and  so  something  more  approaching  to  valid- 
ity. But  we  insist  that  in  their  nature  they  are  the  same  and 
thai  at  most  they  can  differ  only  in  degree.  We,  of  course,  use 
the  term  "induction"  in  this  comparison  as  denoting  a  par- 
ticular kind  of  inference  suggested  by  observed  facts,  and  in- 
cluding more  than  they  contain.  Hypothesis  we  use  as  a  con- 
venient term  for  representing  this  function,  of  whatever  degree, 
in  the  application  of  scientific  method,  and  so  indicating  the 
mental  act  which  must  be  preliminary  to  all  verification,  and 
intermediate  between  the  observation  of  facts  and  the  proof 
of  whatever  guesses,  surmises,  or  probable  inferences  we  may 
make. 

The  term  is  sometimes  compared  with  the  term  theory,  and 
a  distinction  drawn  between  them.  An  hypothesis  is  called  a 
supposition,  meaning  that  it  represents  a  mere  possibility,  and 
a  theory  is  called  a  verified  hypothesis.  But  this  distinction 
is  only  an  arbitrary  one,  except  as  it  implies  the  difference 
between  the  lowest  and  the  higher  degrees  of  probability  in 
various  inferences.  The  fact  is,  also,  that  the  terms  are  often 
used  interchangeably.  We  speak  indifferently  of  the  "Dar- 
winian hypothesis,"  and  the  "  Darwinian  theory  ;  "  the  "  nebu- 
lar hypothesis,"  and  the  "  nebular  theory."  Also  we  say  "  un- 
dulatory  theory  "  of  light  and  sound,  but  never  "  hypothesis," 
although  the  conception  is  precisely  that  of  a  conjecture  or 
supposition  awaiting  satisfactory  verification.  On  the  other 
hand,  the  term  theory  is  sometimes  used  to  comprehend  a 
body  of  ascertained  truth,  or  a  number  of  related  ideas 
brought  together  to  represent  a  system  of  truths.     Thus  we 


SCIENTIFIC  METHOD  345 

say  "the  theory  of  the  universe,"  "the  theory  of  the  solar 
eclipse,"  "the  theory  of  heat,"  "the  theory  of  equations." 
It  is  this  conception  which  has  given  rise  to  the  notion  that 
a  theory  is  a  verified  hypothesis.  But  their  actual  inter- 
changeability  in  many  crucial  and  important  instances,  and 
the  fact  that  they  both  represent  an  inductive  process  hav- 
ing indistinguishable  degrees  of  probability,  are  sufficiently 
cogent  arguments  for  making  them  identical  in  all  essential 
features,  and  so  using  one  of  them  for  describing  the  whole 
process  incident  to  the  attainment  of  new  conceptions  which 
require  verification.  An  hypothesis,  then,  we  regard  as  a  sup- 
position or  inference  from  given  data  to  their  cause,  their 
principle  of  unity,  or  their  identity.  Although  the  term  is 
usually  employed  to  denote  a  very  low  degree  of  probabil- 
ity in  the  supposition,  this  does  not  make  the  act  of  form- 
ing it  different  in  kind  from  more  assured  inductive  infer- 
ences, and  so  we  used  it  merely  to  denote  the  stage  of  thought 
immediately  antecedent  to  verification. 

To  illustrate  :  "When  Franklin  observed  certain  resemblances 
between  lightning  and  electricity,  his  supposition  that  they 
were  identical  was  an  hypothesis.  His  experiment  with  the 
kite  was  of  the  nature  of  a  verification.  The  supposition  that 
the  sun's  heat  is  supplied  from  the  falling  of  meteors  into  the 
sun,  is  an  hypothesis.  So  also  is  the  vulcanic  theory  of  the 
molten  character  of  the  earth's  centre.  Some  such  inference 
or  supposition  from  given  facts  must  first  be  made  before  any 
steps  can  be  taken  to  give  it  greater  probability  or  positive 
proof.  In  some  cases,  of  course,  the  circumstances  give  so 
great  a  probability  to  the  first  inference  that  it  scarcely  re- 
quires verification.  But  the  inference  is  nevertheless  an  hy- 
pothesis, if  we  are  to  take  the  term  in  its  broadest  sense,  to 
denote  what  is  less  than  a  positively  ascertained  certainty. 

The  formation  of  hypotheses  must  be  subject  to  certain  con- 
ditions affecting  the  right  to  make  them.  It  is  important  to 
observe  that  there  can  be  no  absolute  rules  regulating  their 
legitimacy,  because  nothing  can  determine  their  positive  va- 
lidity or  invalidity  except  the  process  of  verification.     Hence 


346  ELEMENTS  OF  LOGIC 

there  must  be  a  wide  range  of  liberty  in  forming  them. 
Nevertheless,  there  are  certain  proximate  rules  regulating 
their  formation  and  determining  the  degree  of  presumption 
in  their  favor,  which  ought  to  be  observed  by  everyone  enter- 
ing into  scientific  study.  Some  of  the  more  important  of 
them  may  be  briefly  discussed. 

(a)  An  hypothesis  should  not  be  inconsistent  with  known  facts 
and  causes,  but  should  be  required  by  new  phenomena  that  are 
not  explained  by  existing  suppositions. 

We  have  no  right  to  introduce  any  new  hypothesis  when  the 
facts,  forces,  or  causes  already  known  are  admittedly  capable 
of  explaining  the  phenomena  under  consideration.  If  there 
be  anything  new  and  unaccounted  for,  there  can  be  no  objec- 
tion to  new  hypotheses  ;  or  if  the  known  causes  account  for 
only  a  part  of  the  effect,  the  new  supposition  is  admissible. 
But  we  must  in  all  such  cases  be  assured  that  the  old  suppo- 
sitions are  not  adequate  to  the  effect,  and  that  there  are  new 
phenomena  to  be  explained.  Thus,  for  example,  if  the  law  of 
supply  and  demand  be  a  fully  accepted  one  as  regulative  of 
prices,  we  should  be  careful  about  introducing  a^y  other  in- 
fluence as  determinative  of  them.  Or,  if  the  material  re- 
sources of  a  country  are  known  to  be  especially  rich  and  fer- 
tile, it  will  not  be  legitimate  to  assume  that  its  prosperity  is 
due  to  government  interference  with  industrial  action,  until  we 
have  shown  phenomena  which  cannot  be  accounted  for  by  any 
other  means.  For  example,  we  often  see  protection  appealed 
to  as  the  cause  of  industrial  wealth  where  the  impression  con- 
veyed is  that  this  influence  is  the  sole  cause  of  it.  But  al- 
though it  may  be  a  factor,  as  long  as  other  causes  are  known 
to  be  active  it  is  illegitimate  to  introduce  protection  as  the 
only  supposition  required  to  account  for  the  effect.  We  must 
first  be  assured  of  an  unexplained  increment.  Or,  to  give  am 
other  illustration,  if  it  be  known  that  petrifactions  actually 
occur,  and  that  animal  and  vegetable  life  may  leave  traces  of 
their  existence  in  deposits  of  sand  and  rock  during  the  pres- 
ent period  of  the  earth's  development,  it  is  unnecessary  to 
suppose  the  existence  of  some  "materia  pinguis,"  fatty  mat- 


SCIENTIFIC  METHOD  3-±7 

ter,  or  "lapidifying  juice"  to  account  for  the  traces  of  fossils 
which  do  not  represent  present  forms  of  organic  life.  It  is 
easier  to  assume  that  nature  is  uniform  where  precisely  the 
same  kinds  of  effects  are  discernible,  and  in  this  case  the 
known  fact  of  death  or  the  extinction  of  life,  in  many  instances, 
has  only  to  be  added  to  the  suppositive  fossil  deposits  in  past 
ages  in  order  to  account  for  the  difference  of  genera  and  species 
which  is  observed.  The  instance  only  proves  that  we  must 
first  exhaust  the  explanatory  power  of  known  causes  before 
resorting  to  new  ones. 

There  are  several  rules  which  may  be  regarded  as  mere  cor- 
ollaries of  the  one  just  discussed,  or  at  least  as  closely  related 
to  it.  Such  as  that  an  hypothesis  should  be  as  simple  and 
as  free  from  complexity  as  possible  ;  or,  that  it  should  not  be 
arbitrary,  but  should  be  relevant  to  the  phenomenon  and  ac- 
cording to  actual  experience.  An  illustration  of  unnecessary 
complexity  is  the  persistence  of  some  astronomers  in  the  theory 
of  epicycles  and  eccentrics  after  the  simpler  hypothesis  of  Co- 
pernicus had  been  proposed.  An  example  of  arbitrary  as- 
sumptions is  that  of  the  Italian  j^hilosopher,  mentioned  by 
Fowler,  who  sought  to  reconcile  the  Aristotelian  and  Platonic 
theory  of  the  rotundity  of  heavenly  bodies,  with  the  supposi- 
tion of  Galileo,  proved  by  observations  of  the  telescope,  that 
the  moon  was  of  a  rough  surface,  by  imagining  that  the  hollow 
parts  were  filled  with  transparent  crystal,  which  would  permit 
the  same  appearances  of  light  and  shadow  as  those  we  observe, 
and  would  observe  without  this  substance.  We  may  reconcile 
facts  by  a  new  hypothesis,  but  we  should  not  undertake  to 
reconcile  theories  or  suppositions  when  the  sinrpler  of  the  two 
may  explain  the  facts  or  cover  all  that  is  known  of  them.  The 
difficulty,  however,  in  all  such  cases  is  to  distinguish  between 
the  facts  and  the  hypotheses,  because  suppositions  which  are 
believed  for  a  long  time,  and  which  rendered  a  large  body  of 
phenomena  intelligible,  come  to  be  considered  as  facts,  and 
so  give  rise  to  attempts  to  reconcile  them  with  new  theo- 
ries apparently  in  conflict  with  them.  What  is  needed  is  a 
criterion    to    distinguish   between   facts   and   hypotheses,    or 


348  ELEMENTS  OF  LOGIC 

verified  and  unverified  theories.  We  turn  next  to  the  second 
rule. 

(b)  An  hypothesis  should  permit  the  application  of  deductive 
reasoning,  or  of  the  inference  to  consequences  ivhich  are  capable 
of  comparison  with  the  results  of  observation. 

This  is  the  first  of  Jevons's  rules,  and  seems  to  have  been 
suggested  to  him  by  the  observations  of  Hobbes  and  Boyle. 
It  differs  from  the  first  rule  we  have  formulated  only  in  the 
requirement  that  the  hypothesis  be  capable  of  deductive  in- 
ferences consistent  with  the  facts  of  observation,  and  so  is  re- 
garded by  Jevons  as  merely  another  form  of  the  rule  which 
he  regards  as  the  sole  test  of  the  legitimacy  of  the  hypothe- 
sis, namely,  conformity  with  observed  facts.  The  simplicity  of 
this  last  mode  of  statement,  however,  is  so  great  that  illegiti- 
mate hypotheses  might  appear  admissible  for  lack  of  the  defi- 
niteness  which  is  necessary  to  make  the  rule  useful. 

The  importance  of  this  rule  is,  that  an  hypothesis  will  not 
be  capable  of  verification  unless  deductive  inferences  can  be 
drawn  from  it,  and  the  results  compared  with  experience  and 
observation.  The  theory  of  the  transparent  crystal  filling  the 
hollow  spaces  of  the  moon's  surface,  which  was  mentioned 
above,  is  of  this  nature.  Perhaps  the  old  theory  of  Phlogis- 
ton is  of  the  same  kind,  since  no  deductions  could  be  made 
from  it.  Jevons  states  the  case  in  the  following  manner  : 
"We  can  only  infer  what  would  happen  under  supposed  con- 
ditions by  applying  the  knowledge  of  nature  we  possess  to 
those  conditions.  Hence,  as  Boscovitch  truly  said,  we  are  to 
understand  by  hypotheses  '  not  fictions  altogether  arbitrary, 
but  suppositions  comformable  to  experience  or  analogy.'  It 
follows  that  every  hypothesis  worthy  of  consideration  must 
suggest  some  likeness,  analogy,  or  common  law,  acting  in  two 
or  more  things.  If  in  order  to  explain  certain  facts  a,  a,  a", 
etc.,  we  invent  a  cause  A,  then  we  must  in  some  degree  ap- 
peal to  experience  as  to  the  mode  in  which  A  will  act.  As  the 
laws  of  nature  are  not  known  to  the  mind  intuitively,  we 
must  point  out  some  other  cause,  B,  which  supplies  the  requi- 
site notions,  and  all  we  do  is  to  invent  a  fourth  term  to  an 


SCIENTIFIC  METHOD  349 

analogy.  As  B  is  to  its  effects  b,  b',  b",  etc.,  so  is  A  to  its 
effects  a,  a',  a",  etc.  When  we  attempt  to  explain  the  passage 
of  light  and  heat  radiations  through  space  unoccupied  by 
matter,  we  imagine  the  existence  of  the  so-called  ether.  But 
if  this  ether  were  wholly  different  from  anything  else  known 
to  us,  we  should  in  vain  try  to  reason  about  it.  We  must  ap- 
ply to  it  at  least  the  laws  of  motion,  that  is,  we  must  so  far 
liken  it  to  matter.  And  as,  when  applying  those  laws  to  the 
elastic  medium  air,  we  are  able  to  infer  the  phenomena  of 
sound,  so  by  arguing  in  a  similar  manner  concerning  ether,  we 
are  able  to  infer  the  existence  of  light  phenomena  correspond- 
ing to  what  do  occur.  All  that  we  do  is  to  take  an  elastic 
substance,  increase  its  elasticity  immensely,  and  denude  it  of 
gravity  and  some  other  properties  of  matter,  but  we  must  re- 
tain sufficient  likeness  to  matter  to  allow  of  deductive  calcula- 
tions." 

These  two  main  rules  are  sufficient  for  all  practical  pur- 
poses. Most  other  rales  are  either  corollaries  of  the  second, 
or  combinations  of  this  and  the  corollaries  of  the  first  rule. 
Hence  there  is  no  need  of  multiplying  them  unnecessarily. 

It  is  important  to  remark,  however,  that  conformity  to 
these  rules  does  not  verify  or  prove  an  hypothesis.  It  only 
shows  their  right  to  be  entertained  as  representing  a  possi- 
bility or  probability.  They  are  legitimately  made  when  they 
explain  a  given  number  of  facts,  but  may  be  set  aside  by  a 
better  one  which  supersedes  the  place  of  the  old  one  on  ac- 
count of  superior  simplicity  and  comprehension,  or  superior 
deductive  capacity  in  connection  with  observed  facts. 

Hypotheses  are  sometimes  advanced  for  the  purpose  of  de- 
ductions which  shall  be  a  reductio  ad  absurdum.  These  cor- 
respond to  the  negative  proof  of  deductive  methods,  and  as- 
sume a  positive  hypothesis  already  in  mind,  and  in  this  way 
conform  to  the  rules  we  have  laid  down. 

2d.  Verification. — Verification  is  the  process  of  testing 
hypotheses  and  inferences  to  see  whether  they  have  been  cor- 
rect or  not.  After  a  supposition  has  been  made,  or  an  in- 
ference drawn  that   something  is  possibly  or  probably  true, 


350  ELEMENTS  OF  LOGIO 

the  mind  naturally  looks  for  other  evidence  to  substantiate 
them.  This  evidence  may  come  in  the  repeated  occurrence 
of  the  original  phenomenon,  showing  that  it  is  probably  not  a 
mere  accident  of  circumstances,  or  it  may  come  in  its  being  a 
necessary  deduction  from  some  known  fact  or  law,  but  was 
not  perceived  until  the  event  was  better  known.  The  verifi- 
cation, then,  is  intended  to  give  greater  assurance  than  the 
hypothesis  can  be  supposed  to  have  on  the  first  occasion  of 
its  suggestion  to  the  mind.  When  Newton  first  thought  of 
the  law  of  universal  gravitation,  he  was  not  content  with  its 
power  to  explain  the  single  phenomenon  which  suggested  it, 
but  he  saw  that  certain  other  facts  must  follow  from  it,  or 
must  inevitably  be  associated  with  it.  He  therefore  set  about 
a  mathematical  calculation  to  see  if  the  result  coincided  with 
what  ought  to  occur  in  the  case,  and  seeing  that  it  did  not,  he 
gave  up  his  theory  until  new  data,  some  ten  years  later,  were 
discovered  regarding  the  true  distance  of  the  moon  from  the 
earth.  He  then  resumed  his  calculations,  and  found  the  re- 
sult to  coincide  with  his  hypothesis,  and  this  was  regarded  as 
a  verification  of  it.  And  so,  when  the  hypothesis  that  all 
gases  might  be  compressed  into  liquids  or  solids  was  ad- 
vanced, it  was  at  least  a  partial  verification  of  it  to  have  suc- 
ceeded in  compressing  hydrogen  into  a  liquid  under  a  low 
temperature  and  high  pressure. 

The  process  of  verification  may  be  of  two  kinds.  The  first 
is  that  which  increases  the  probability  of  an  hypothesis  or  in- 
ference by  the  four  inductive  methods  ;  the  second  is  that  which 
establishes  the  certitude  of  it  by  deductive  or  by  observational 
principles.  The  one  insensibly  passes  into  the  other,  as  we 
shall  see  when  considering  certain  peculiarities  of  the  method 
of  difference  ;  or  the  two  may  be  combined  in  such  a  way  that 
no  absolute  line  of  distinction  can  be  maintained,  and  we  can 
only  rely  upon  extreme  instances  of  their  application  for  mak- 
ing the  distinction  at  all.  The  manner  in  which  the  two 
methods  may  be  connected,  consciously  or  unconsciously,  will 
be  remarked  in  its  place.  We  must  take  the  several  forms  of 
verification  according  to  their  simplicity,  and  not  according  to 


SCIENTIFIC  METHOD  351 

the  degree  of  certitude  supplied  by  them.  These  we  reduce  to 
four  :  Observation,  Experiment,  Inductive  Methods,  aud  Deduc- 
tion, and  shall  treat  them  in  that  order. 

1.  Observation. — Observation  in  verification  is  the  same 
process  that  it  is  in  acquisition,  only  it  is  applied  with  some- 
what different  presumptions.  When  an  inference  or  hypothe- 
sis has  once  been  made,  observation  may  come  in  to  ascertain 
whether  it  is  correct  or  not,  and  according  as  the  expectation 
is  realized  or  not,  will  the  verification  be.  Thus,  when  the 
Copernican  theory  of  planetary  motion  around  the  sun  was 
proposed  and  the  explanation  of  the  phases  of  the  moon  was 
accepted,  it  was  argued  that  Mercury  and  Venus  ought  to  ex- 
hibit similar  phases.  This  was  admitted  by  the  advocates  of 
the  theory,  and  when  Galileo  turned  his  telescope  upon  them 
the  inference  was  verified  by  the  discovery  of  these  phases. 
Here  the  observation  of  a  fact  was  the  condition  of  proving 
the  hypothesis.  Whenever  the  inference  concerns  the  exist- 
ence of  some  physical  fact  beyond  the  control  or  production 
of  experiment,  observation  is  the  only  resource  for  verification, 
and  the  assumption  made  is,  that  the  realization  of  expectation 
cannot  be  due  to  chance.  The  effect  of  finding  the  inference 
turn  out  true  is  like  that  of  seeing  a  prediction  realized.  It  is 
a  proof  of  the  assumptions  on  which  it  is  made.  If  I  have  in- 
ferred from  observations  in  a  particular  region  that  ague  is 
due  to  miasma  fi-om  swampy  soil,  it  will  be  something  of  a 
verification  of  my  hypothesis  to  observe  that  the  same  disease  is 
associated  with  like  conditions  in  other  places,  and  my  certi- 
tude regarding  it  will  depend  upon  the  number  of  my  subse- 
quent observations,  and  the  varied  circumstances  under  which 
the  disease  occurs. 

The  certainty  which  verification  by  observation  gives  varies 
with  the  nature  of  the  conditions  under  which  it  is  made.  If 
we  are  assured  of  the  simple  and  isolated  nature  of  the  con- 
ditions, it  may  amount  to  positive  proof.  But  if  the  combina- 
tion of  the  conditions  is  great,  the  observations  may  only  in- 
crease the  probability  of  our  hypothesis,  and  this  varies 
between  the  lowest  and  the  highest  degree,  which  last  in  most 


352  ELEMENTS  OF  LOGIC 

cases  cannot  be  distinguished  from  demonstration  or  certainty, 
so  far  as  the  feelings  of  the  mind  are  concerned. 

2.  Experiment. — As  already  remarked,  experiment  is  only 
a  modified  method  of  observation.  But  in  the  process  of  veri- 
fication it  is  much  more  valuable  than  observation.  It  is  the 
artificial  reproduction  of  phenomena  under  conditions  that 
isolate  them  more  successfully  than  we  can  expect  to  find 
them,  or  at  least  be  assured  of,  in  nature.  Besides,  it  enables 
us  to  vary  the  circumstances  under  which  the  effect  may 
occur.  Such  was  the  effect  of  Sir  David  Brewster's  experi- 
ment showing  that  iridescence  was  due  to  the  form  and  not 
the  nature  of  a  substance.  By  taking  a  wax  impression  of 
mother-of-pearl,  as  before  mentioned,  he  produced  the  effect 
where  there  could  be  no  condition  affecting  the  result  but  the 
form  of  the  surface,  and  hence  whatever  doubt  might  have  ex- 
isted in  regard  to  the  observation  of  natural  instances  was 
here  removed  by  a  decisive  experiment.  In  observing  the  oc- 
currence of  phenomena  in  nature  there  may  be  so  many  con- 
ditions associated  with  their  production  that  a  doubt  may 
often  exist  regarding  the  conclusiveness  of  the  verification  by 
observation,  because  the  effect  may  be  due  to  other  than  the 
inferred  cause  which  has  not  yet  been  separated  from  the 
others.  But  experiment  eliminates  many  possible  conditions 
of  the  phenomenon,  and  so  gives  greater  assurance  to  our  in- 
ferences. This  certitude  may  vary  from  all  degrees  of  prob- 
ability to  positive  certitude,  the  degree  depending  wholly 
upon  the  amount  of  isolation  effected  in  the  conditions  of  the 
phenomenon.  It  is  the  great  means  of  verifying  conjectures 
in  the  physical  sciences,  and  has  more  force  than  any  other 
method. 

It  should  be  remarked,  however,  that  both  observation  and 
experiment,  or  modes  of  verification,  are  very  likely  to  be  as- 
sociated either  with  the  inductive  methods  of  Agreement  and 
Difference,  or  with  inductive  principles,  as  verifying  instru- 
ments. Indeed,  it  may  be  said  that  observation  can  hardly  be 
separated  from  them.  But  the  two  processes  will  verify  an 
hypothesis  in  proportion  to  the  extent  of  their  connection 


SCIENTIFIC  METHOD  353 

with  the  above-mentioned  methods  and  principles.  If  obser- 
vation enlarges  the  number  of  instances  illustrating  the 
method  of  agreement,  it  increases  the  probability  of  the  infer- 
ence, and  if  it  detects  a  case  or  several  cases  of  difference, -the 
probability  may  amount  to  a  proof.  Experiment  will  do  the 
same.  It  will  enlarge  the  area  of  application  for  the  method 
of  agreement,  and  more  easily  effects  an  application  of  the 
method  of  difference,  so  that  its  verifying  power  is  greater 
than  that  of  observation.  And,  of  course,  if  any  deductive  as- 
sumptions, or  known  facts  and  principles,  are  illustrated  by 
the  results  of  the  two  processes,  the  verification  becomes  a  de- 
monstrated certainty.  Usually  some  deductive  assumption  is 
made,  whether  of  a  probable  or  an  assured  truth,  before  ob- 
servation and  experiment  are  applied.  Hence  we  must  always 
calculate  whether  our  assurance  or  verification  is  due  more  to 
the  methods  of  observation  and  experiment  than  to  the  realiza- 
tion of  assumptions  already  made  under  the  inspiration  of 
agreement  and  difference  or  deductive  truths. 

3.  Inductive  Methods. — We  have  already  explained  the 
methods  of  agreement  and  difference  as  principles  of  induction, 
or  grounds  of  the  inductive  inference  and  hypothesis.  It  re- 
mains to  show  that  the  same  principles  may  be  used  for  veri- 
fication, in  that  they  may  be  the  means  or  conditions  of 
increasing  the  probability  of  our  inductive  reasoning.  They 
renew  the  inference  under  circumstances  which  increase  its  co- 
gency and  probability.  They  are,  however,  quite  as  frequently 
the  source  of  the  inductive  inference  or  hypothesis  as  the 
means  of  verification.  They  act  as  verifying  agencies,  when 
we  can  assume  that  the  re-occurrence  of  a  phenomenon  in 
accordance  with  the  conditions  involved  in  these  methods  ful- 
fils the  expectation  expressed  by  our  hypothesis,  and  so  the 
original  inference  is  confirmed  and  strengthened  by  the  mul- 
tiplication of  the  instances  involving  the  effect.  The  verifica- 
tion consists  more  in  the  number  or  quantity  than  in  the 
nature  or  quality  of  the  instances,  because  the  fact  of  fre- 
quency is  a  better  indication  of  a  law  of  nature.  Although  we 
speak  a  great  deal  of  the  uniformity  of  nature,  there  is  as  much 


354  ELEMENTS  OF  LOGIC 

variety  at  the  same  time,  and  it  is  this  variety  that  acts  as  a 
disturbing  factor  to  our  expectations  and  calculations.  Be- 
sides, this  variety  and  diversity  indicate  a  constant  change  of 
causal  agencies,  so  that  we  may  never  be  assured  that  an  event 
will  have  the  same  connections  a  second  time.  But  if  these 
connections  are  repeated  under  a  variety  of  circumstances  and 
other  changes,  the  supposition  suggested  by  a  peculiar  inci- 
dent will  be  confirmed  because  of  an  approximation  thus  made 
to  a  law  of  nature.  This  is  the  Method  of  Agreement.  But 
if  with  the  isolation  of  a  particular  cause  we  discover  a  simul- 
taneous isolation  of  a  given  phenomenon,  we  may  infer  their 
causal  connection.  But  the  verification  of  this  inference  by 
the  Method  of  Difference,  as  it  was  suggested  by  the  same 
method,  may  come  in  one  of  two  ways  yet  to  be  noticed.  We 
take  up  the  methods  separately. 

(a)  Method  of  Agreement. — This  has  been  sufficiently  de- 
fined. To  understand  its  application  as  a  source  of  verifica- 
tion, we  have  only  to  recall  the  illustration  of  iridescence, 
where  the  common  fact  was  this  phenomenon  in  connection 
with  a  variety  of  substances.  By  increasing  this  variety  of 
conditions  and  retaining  the  common  fact  of  iridescence,  I 
increase  the  probability  that  the  cause,  which  was  supposed 
after  a  few  observations  to  be  the  form  of  surface  on  the 
bodies,  is  the  true  one.  The  highest  degree  of  probability  in 
this  case  will  be  reached  where  the  observations  pass  over  to 
the  method  of  difference,  which  we  shall  notice  presently. 

Again,  I  find  that  after  taking  a  particular  kind  of  food  I 
am  ill.  But  I  have  taken  it  along  with  other  kinds  at  the 
same  time,  and  my  conjecture  that  the  effect  may  be  due  to 
the  particular  kind  supposed  will  be  very  weak,  until  I  have 
repeated  the  circumstances  frequently  enough  to  confirm  my 
belief  in  the  supposed  cause.  This  is  greatly  strengthened  by 
changes  of  general  condition— climate,  temperature,  air,  and 
food — which  show  that  this  presence  or  absence  are  not  mater- 
ial to  the  result.  Here  again  is  an  instance  where  the  hypo- 
thesis was  first  suggested  by  the  principle  of  agreement,  and 
then  verified  or  confirmed  by  the  same,  on  an  extension  of  the 


SCIENTIFIC  METHOD  355 

instances  to  which  it  was  applicable  ;  and  it  is  also  noticeable 
that  the  verification  increases  in  probability  or  cogency  with 
the  approximation  of  the  case  to  one  involving  the  method  of 
difference.  This  approximation  to  the  method  of  difference 
consists  in  the  constant  elimination  of  conditions  which  might 
possibly  be  the  cause  of  the  phenomenon  in  question  ;  that  is, 
the  elimination  of  what  appears  by  this  separation  not  to  be 
the  cause,  until  the  phenomenon  and  its  causes  are  left  in 
complete  isolation,  when  the  inference  becomes  distinctly  veri- 
fied by  the  fact. 

(6)  Method  of  Difference. — Mr.  Mill  states  the  Canon  of 
Difference  in  the  following  language  :  "  If  an  instance  in  which 
the  phenomenon  under  investigation  occurs,  and  an  instance 
in  which  it  does  not  occur,  have  every  circumstance  in  com- 
mon save  one,  that  one  occurring  only  in  the  former  ;  the  cir- 
cumstance in  which  alone  the  two  instances  differ  is  the  effect 
or  the  cause,  or  an  indispensable  part  of  the  cause,  of  the 
phenomenon." 

A  clear  conception  of  it  may  be  afforded  by  comparing  it 
with  the  method  of  agreement.  In  the  method  of  agreement, 
everything  may  vary  except  the  phenomena  in  question  ;  in  the 
method  of  difference,  nothing  may  vary  ;  that  is,  everything 
may  be  the  same,  except  the  phenomena  in  question.  In  the 
latter  case  the  probability  of  the  inference  is  always  regarded 
as  greater  than  in  the  former.  The  reason  for  this  is  that  with 
the  method  of  agreement  it  is  assumed  that  several  causes  may 
produce  the  same  effect,  and  hence  the  probability  that  any 
one  of  them  is  the  true  one  is  diminished.  Besides,  we  are 
arguing  from  effect  to  cause.  On  the  other  hand,  with  the 
method  of  difference,  when  everything  is  common  except  the 
phenomena  in  question,  we  have  instances  in  which  causes  are 
present  and  yet  the  effect  to  be  accounted  for  does  not  follow, 
but  something  else  connected  with  another  isolated  circum- 
stance. If  these  common  qualities  were  causes  of  the  phe- 
nomenon, it  must  accompany  them  ;  but  as  it  does  not  accom- 
pany them  they  cannot  be  its  causes.  Hence,  since  the  common 
qualities  are  not  the  causes,  we  have  no  other  choice  but  to 


356  ELEMENTS  OF  LOGIC 

accept  the  antecedent  which  has  been  isolated  with  the  phe- 
nomenon as  its  cause.  The  probability  is  here  proportioned 
to  the  assurance  we  feel  that  the  phenomena  have  really  been 
isolated.  Besides,  a  part  of  the  argument  is  from  cause  to 
effect,  and  deductive.  That  is  deductive  which  proves  what  is 
not  the  cause,  and  the  remainder,  even  if  regarded  as  induc- 
tive, has  its  assurance  based  upon  the  probabilities  of  complete 
isolation  in  the  phenomena  concerned. 

A  passage  from  Jevons  will  furnish  illustrations  of  the 
method  of  difference.  "  Thus  we  can  clearly  prove  that  friction 
is  one  cause  of  heat,  because  when  two  sticks  are  rubbed  to- 
gether they  become  heated  ;  when  not  rubbed  together  they 
do  not  become  heated.  Sir  Humphry  Davy  showed  that  even 
two  pieces  of  ice  when  rubbed  together  in  a  vacuum  produce 
heat,  as  shown  by  their  melting,  and  thus  completely  demon- 
strate that  the  friction  is  the  source  and  cause  of  the  heat.  We 
prove  that  air  is  the  cause  of  sound  being  communicated  to 
our  ears  by  striking  a  bell  in  the  receiver  of  an  air-pump,  as 
Hawksbee  first  did  in  1715,  and  then  observing  when  the  re- 
ceiver is  full  of  air  we  hear  the  bell ;  when  it  contains  little  or 
no  air  we  do  not  hear  the  bell.  We  learn  that  sodium,  or  any 
of  its  compounds,  produces  a  spectrum  having  a  bright-yellow 
double  line,  by  noticing  that  there  is  no  such  line  in  the  spec- 
trum of  light  when  sodium  is  not  present,  but  that  if  the  small- 
est quantity  of  sodium  be  thrown  into  the  flame  or  other  source 
of  light,  the  bright  yellow  line  instantly  appears.  Oxygen  is 
the  cause  of  respiration  and  life,  because  if  an  animal  be  put 
into  a  jar  full  of  atmospheric  air,  from  which  the  oxygen  has 
been  withdrawn,  it  soon  becomes  suffocated." 

These  are  instances  where  the  method  of  difference  is  applied, 
and  it  will  be  apparent  on  examination  that  the  inference 
turns  upon  the  elimination  of  the  two  phenomena  with  each 
other  while  all  else  remains  the  same.  But  now  it  requires  to 
be  shown  how  the  method  is  one  of  verification  as  well  as  of 
suggesting  the  inductive  inference.  The  inductive  inference 
is  from  the  fact  to  its  cause  ;  the  verification  is  an  increase 
of  its  probability  or  its  proof.     How  does  the  method  of  dif- 


SCIENTIFIC  METHOD  357 

ference  effect  this  at  the  same  time  that  it  is  the  source  of 
the  inference  ? 

In  answer  to  this  question  it  is  proper  to  remark  first,  that 
some  logicians,  notably  Mr.  Venn,  regard  the  method  of  differ- 
ence as  properly  one  of  verification,  and  not  of  suggestion. 
But  this  mere  expression  of  opinion  is  not  a  proof  of  it,  nor 
have  adequate  reasons  been  given  for  it.  The  first  inci- 
dent of  its  verifying  power,  however,  would  be  the  repetition 
of  the  observations  and  experiments  involving  its  application, 
so  as  to  show  that  the  phenomena  concerned  were  not  acci- 
dental, and  in  this  way  the  inference  would  be  strengthened. 
But  usually  the  inference  seems  so  cogent  that  it  requires  no 
such  repetition  to  reinforce  it.  The  mind  seems  convinced 
positively  by  a  single  instance.  The  inference  and  its  verifi- 
cation seem  to  occur  at  once.  The  reason  for  this  is  the  exist- 
ence of  certain  assumptions  that  are  an  invariable  part  of  the 
method  of  difference.  In  the  first  place,  there  is  the  deduc- 
tive argument  involved,  which  we  have  observed,  and  which 
shows  what  is  not  the  cause.  The  separation  of  the  common 
qualities  from  the  phenomenon  to  be  explained  settles  this 
fact.  In  the  second  place,  the  assumption  made  whenever  the 
method  of  difference  is  strictly  applied  and  which  is,  that  there 
are  no  other  than  the  isolated  circumstances  to  be  taken  into  ac- 
count, leaves  no  room  for  doubt,  and  verifies  the  case  by  an- 
other implied  deductive  syllogism  with  this  assumption  as  its 
major  premise.  Of  course  this  may  be  doubtful,  but  this  does 
not  alter  the  form  of  the  argument  or  the  method  of  verification, 
and  hence  we  see  that  the  whole  certainty,  or  the  degree  of 
probability  attaching  to  the  inference,  is  determined  by  the  as- 
surance we  feel  that  there  are  no  other  circumstances  to  be 
considered  in  the  case.  These  are  the  verifying  agencies  in 
the  method  of  difference,  and  they  are  probably  associated 
with  it  in  all  its  forms  and  applications. 

There  are  three  other  methods  which  are  sinrply  modifica- 
tions of  one  or  the  other  of  the  methods  just  considered,  and 
which  may  be  briefly  considered. 

(c)  Method  of  Concomitant  Variations. — This  is  the  method  of 


35S  ELEMENTS  OF  LOGIC 

difference  applied  in  circumstances  where  the  variations  of  one 
phenomenon  are  constant  and  simultaneous  with  another. 
Thus,  all  other  things  being  equal  or  remaining  the  same,  the 
variations  of  the  altitude  of  the  mercury  in  a  thermometer 
along  with  a  corresponding  variation  of  temperature,  lead  to 
the  inference  that  the  temperature  is  the  cause,  and  the  veri- 
fication by  the  same  method  is  as  before.  "We  may  repeat  the 
observation  or  experiment,  or  use  the  reasoning  which  is  in- 
cident to  the  method  of  difference  in  order  to  add  more  prob- 
ability to  our  inference.  In  so  far  as  the  method  is  one  of  dif- 
ference it  represents  more  of  verification  than  suggestion  or 
acquisition. 

(d)  Method  of  Residues. — This  is  again  another  application 
of  the  method  of  difference.  If  we  subtract  or  eliminate  from 
any  group  of  phenomena  what  we  know  is  due  to  a  given 
cause,  the  residue  will  be  the  effect  of  the  remaining  condi- 
tions. Thus,  if  we  suspect  the  impurity  of  a  mass  of  gold,  we 
have  only  to  determine  its  specific  gravity  and  conrpare  this 
with  what  we  know  of  the  specific  gravity  of  pure  gold.  The 
existence  of  a  difference  proves  the  case.  We  can  thus  deter- 
mine what  influence  the  sun  has  on  the  tides  by  subtracting 
the  influence  of  the  moon  from  the  total  effect. 

(e)  Joint  Method  of  Agreement  and  Difference. — "This  meth- 
od consists  in  a  double  employment  of  the  method  of  agree- 
ment and  a  comparison  of  the  results  thus  obtained,  the  com- 
parison assimilating  it  to  the  method  of  difference.  We,  first 
of  all,  compare  cases  in  which  the  phenomenon  occurs,  and, 
so  far  as  we  can  ascertain,  find  them  to  agree  in  the  posses- 
sion of  only  one  other  circumstance.  But,  though  we  may 
not  be  justified  in  regarding  this  inference  as  certain,  we  may 
increase  our  assurance  by  proceeding  to  compare  cases  in 
which  the  phenomenon  does  not  occur,"  The  assurance  may 
not  be  complete  in  this  case  as  in  the  method  of  difference, 
but  it  is  greater  than  in  the  method  of  agreement.  We  quote 
Fowler's  illustration  of  the  method  : 

"  A  very  thin  sheet  of  light  proceeding  from  incandescent 
hydrogen  is  passed  through  a  prism,  and  it  is  invariably  found 


SCIENTIFIC  METHOD  359 

(with  one  exception)  that  in  the  spectrum  thus  obtained  there 
are,  in  proportion  to  the  intensity  of  the  light,  one,  two,  or 
more  bright  lines  occupying  precisely  the  same  relative  posi- 
tion. Moreover,  very  thin  sheets  of  white  light  proceeding 
from  various  incandescent  substances  are  passed  through  in- 
candescent hydrogen,  and  the  emergent  light  is  then  separated 
into  its  constituent  elements  by  a  prism.  In  the  spectra  thus 
obtained  it  is  found  that  there  are  invariably  (with  the  above- 
mentioned  exception)  dark  lines  occupying  exactly  the  same 
positions  in  the  spectrum  as  the  lines  above  mentioned.  Hence 
it  is  inferred,  by  the  method  of  agreement,  that  a  sheet  of  light, 
whether  it  proceed  directly  from  incandescent  hydrogen  itself, 
or  be  transmitted  through  it  from  some  other  incandescent 
substance,  will  invariably  (allowing  for  the  exceptional  case) 
produce  these  Hues.  But,  if  we  try  the  same  experiments  with 
any  other  element  than  incandescent  hydrogen,  although  we 
may  obtain  bright  or  dark  lines,  we  never  find  tbese  lines  oc- 
cupying the  same  positions  in  the  spectrum  as  the  lines  in 
question." 

The  principles  of  verification  involved  in  the  joint  method 
of  agreement  and  difference  are  those  of  the  two  methods 
themselves,  and  require  no  illustration  at  length.  We  turn  to 
the  last  and  most  conclusive  methods  of  verifying  hypotheses. 

4.  Deduction. — This  method  of  verification  is  simply  the 
application  of  deductive  reasoning  to  show  that  the  conclu- 
sion of  the  inductive  process  may  be  a  necessary  deduction 
from  some  other  known  fact,  law,  or  principle.  We  have  seen 
what  place  it  has  in  the  method  of  difference,  but  it  may  be 
applied  in  many  instances  to  the  inferences  obtained  by  the 
method  of  agreement,  and  it  may  possibly  be  a  contingent 
factor  in  all  forms  of  verification.  But  we  shall  not  insist 
upon  this  possibility.  It  suffices  to  find  some  cases  where  the 
proof  of  an  inference  originating  inductively  may  become  de- 
ductive and  so  absolutely  assuring  it. 

This  deductive  method  of  verification  merely  consists  in 
comparing  an  hypothesis  or  inductive  inference  with  a  known 
truth,  and  ascertaining  whether  they  agree  or  not.    If  they  do 


360  ELEMENTS  OF  LOGIC 

not  agree,  but  contradict  each  other,  we  have  positive  proof  by 
the  principles  of  opposition  that  the  inference  is  false.  If  they 
do  agree,  and  the  inference  bears  all  the  marks  of  being  identi- 
cal with  the  known  truth,  it  must  be  time.  Whenever  it  is 
possible  the  method  is  always  resorted  to.  The  certitude  it 
possesses  depends  wholly  upon  the  certitude  of  the  general 
principle  by  which  the  particular  inference  is  tried.  But  in 
all  cases  where  a  conclusion  by  induction  can  be  confirmed  by 
its  agreement  with  some  other  known  truth  or  principle,  its 
probability  is  very  greatly  increased,  and  if  the  incidents  are 
of  the  proper  kind  the  verification  may  be  demonstrative. 

The  most  conclusive  illustrations  of  verification  by  the  de- 
ductive method  are  those  where  the  hypothesis  accords  with 
the  results  of  mathematical  calculation.  Thus  the  conformity 
of  Newton's  theory  of  gravitation  with  the  mathematical  cal- 
culations made  in  regard  to  planetary  motion  was  universally 
taken  as  a  proof  of  the  theory.  Then  again,  Newton's  Bi- 
nomial Theorem  was  at  first  an  inductive  inference  from  a 
few  observed  instances,  and  it  was  afterward  proved  deduc- 
tively by  later  mathematicians  ;  that  is,  it  was  shown  by  ad- 
mitted mathematical  principles  that  the  co-efficients  and  ex- 
ponents inductively  inferred  by  Newton  must  be  the  ones 
assumed.  Galileo  inferred  from  some  observations  and  ex- 
periments that  the  area  circumscribed  by  a  cycloidal  curve  is 
three  times  that  of  the  generating  circle  or  wheel.  But  he 
had  no  means  of  proving  it,  although  he  tried  to  do  so  exper- 
imentally "  by  cutting  out  cycloids  in  pasteboard,  and  then 
comparing  the  areas  of  the  curve  and  the  generating  circle  by 
weighing  them.  In  every  trial  the  curve  seemed  to  be  rather 
less  than  three  times  the  circle,"  so  that  he  began  to  suspect 
the  correctness  of  his  supposition.  But  afterward  Torricelli 
showed  by  mathematical  calculations  that  this  ratio  m  ust  be 
true. 

The  method  has  applications  o\itside  of  mathematics.  If 
we  know  any  general  principle  we  may  compare  it  with  an  as- 
sumed or  inferred  fact  and  test  its  validity.  Thus  if  I  infer 
from  a  few  observations  of  the  fact  of  potatoes  driving  out 


SCIENTIFIC  METHOD  361 

wheat  as  a  food,  that  the  cheaper  food  has  a  tendency  to  sup- 
plant the  more  costly,  I  may  deduce  this  fact  from  Gresham's 
law  regarding  the  influence  of  cheaper  upon  better  ciu-rencies, 
when  I  know  or  assume,  of  course,  that  money  is  only  a  com- 
modity like  all  other  things.  Mr.  Fowler  mentions  two  in- 
stances of  the  combination  of  induction  and  deduction  which 
are  to  the  point. 

"  In  the  science  of  Political  Economy,  Ricardo's  theory  of 
rent,  when  stated  in  the  slightly  modified  form  that  'the  rent 
of  land  represents  the  pecuniary  value  of  the  advantages 
which  such  land  possesses  over  the  least  valuable  land  in  cul- 
tivation,' is  an  easy  deduction  from  two  principles  which  are 
supplied  by  every  one's  experience  ;  namely,  (1)  that  land 
varies  in  value,  and  (2)  that  there  is  some  land  either  so  bad 
or  so  disadvantageously  situated  as  to  be  not  worth  the  culti- 
vating." 

Professor  Caimes'  work  on  the  Slave  Power  furnishes  a  re- 
markable example  of  the  successful  application  of  the  deduc- 
tive method  to  the  determination  of  economical  questions. 
"  The  economical  effects  of  slavery  are  thus  traced.  We  learn 
from  observation  and  induction  that  slave  labor  is  subject  to 
certain  characteristic  defects  :  it  is  given  reluctantly  ;  it  is  un- 
skilful ;  and,  lastly,  it  is  wanting  in  versatility.  As  a  conse- 
quence of  these  characteristics,  it  can  only  be  employed  with 
profit  when  it  is  possible  to  organize  it  on  a  large  scale.  It  re- 
quires constant  supervision,  and  this  for  small  numbers  or  for 
dispersed  workmen  would  be  too  costly  to  be  remunerative. 
The  slaves  must,  consequently,  be  worked  in  large  gangs. 
Now,  there  are  only  four  products  which  repay  this  mode  of 
cultivation,  namely,  cotton,  sugar,  tobacco,  and  rice.  Hence  a 
country  in  which  slave  labor  prevails  is  practically  restricted 
to  these  four  products,  for  it  is  another  characteristic  of  slave 
labor,  under  its  modern  form,  that  free  labor  cannot  exist  side 
by  side  with  it.  But  besides  restricting  cultivation  to  these 
four  products,  some  or  all  of  which  have  a  peculiar  tendency 
to  exhaust  the  soil,  slave  labor,  from  its  want  of  versatility, 
imposes  a  still  further  restriction.      '  The  difficulty  of  teaching 


302  ELEMENTS  OF  LOGIC 

the  slave  anything  is  so  great — the  result  of  compulsory  igno- 
rance in  which  he  is  kept,  combined  with  want  of  intelligent 
interest  in  his  work— that  the  only  chance  of  rendering  his 
labor  profitable  is,  when  he  has  once  learned  a  lesson,  to  keep 
him  to  that  lesson  for  life.  Accordingly,  where  agricultural 
operations  are  carried  on  by  slaves,  the  business  of  each  gang 
is  always  restricted  to  the  raising  of  a  single  product.  "What- 
ever crop  be  best  suited  to  the  character  of  the  soil  and  the 
nature  of  slave  industry,  whether  cotton,  tobacco,  sugar,  or 
rice,  that  crop  is  cultivated,  and  that  crop  only.  Rotation  of 
crops  is  thus  precluded  by  the  conditions  of  the  case.  The 
soil  is  tasked  again  and  again  to  yield  the  same  product,  and 
the  inevitable  result  follows.  After  a  short  series  of  years  its 
fertility  is  completely  exhausted,  the  planter  abandons  the 
ground  which  he  has  rendered  worthless,  and  passes  on  to  seek 
in  new  soils  for  that  fertility  under  which  alone  the  agencies 
at  his  disposal  can  be  profitably  employed.'  Thus  from  the 
characteristics  of  slave  labor  may  be  deduced  the  economical 
effect  of  exhaustion  of  the  soil  on  which  it  prevails,  and  the 
consequent  necessity  of  constantly  seeking  to  extend  the  area 
of  cultivation.  From  the  peculiar  character  of  the  crops  which 
can  alone  be  successfully  raised  by  slave  labor  may  be  explained 
the  former  prevalence  of  slavery  in  the  southern,  and  its  ab- 
sence in  the  northern  States  of  the  American  Union  ;  and  from 
the  necessity  of  constantly  seeking  fertile  virgin  soil  for  the 
employment  of  slave  labor  may  be  explained  the  former  policy 
of  the  southern  States,  which  was  invariably  endeavoring  to 
bring  newly  constituted  States  under  the  dominion  of  slave 
institutions." 

The  confirmation  of  an  inductive  hypothesis  by  deductive 
reasoning  from  accepted  principles  is  very  common,  and  it 
varies  in  degrees  of  assurance  according  to  the  assurance  we 
feel  about  the  principles  or  premises  upon  which  it  depends. 
As  we  have  already  explained,  it  may  still  be  deductive,  al- 
though the  premise  may  only  be  probably  true,  provided  the 
verification  is  reasoned  out  deductively.  But  we  require  in 
such  cases  to  be  cautious  about  mistaking  another  instance  of 


SCIENTIFIC  METHOD  363 

comparison  by  agreement  for  a  deductive  premise.  We  may 
take  as  proof  in  some  cases  merely  another  instance  of  the 
same  kind  and  so  only  increase  the  probability  of  our  inference 
by  enlarging  the  area  of  the  induction.  Hence  we  must  be 
careful,  in  attempts  at  deductive  verification,  to  secure  a 
principle  or  a  group  of  facts  under  which  the  particular  case 
or  cases  involved  in  the  hypothesis  may  be  placed.  The  in- 
stances, however,  where  this  method  is  applicable  are  mostly 
in  Physics,  or  Mechanics,  Astronomy,  and  the  Mathematics — 
physical  sciences  in  general.  They  are  more  rare  in  what  may 
be  called  the  moral  sciences. 

IV.  FALLACIES  AND  ERRORS  IN  INDUCTIONS.—  Errors 
in  application  of  scientific  method,  when  it  is  purely  inductive, 
are  not  so  easily  detected  as  in  deduction.  There  are  no 
fixed  rules  for  determining  the  degrees  of  probability  attaching 
to  inductive  inferences,  and  hence  the  errors  in  them  are 
mostly  discoverable  only  after  some  contradictory  truth  has 
been  proved.  But  it  is  known  that  we  are  liable  to  mistakes 
in  the  process,  and  hence,  in  order  to  avoid  them  we  are  re- 
quired to  observe  certain  conditions  as  precautions  against 
error,  or  to  keep  in  mind  the  sources'  of  error  that  may  appear 
after  the  attempts  at  verification.  These  sources  of  error  may 
be  divided  into  two  kinds  :  first,  Errors  of  Observation,  and 
second,  Fallacies  of  Inductive  Inference.  The  latter  is  an 
error  connected  with  reasoning. 

I  st.  Errors  of  Observation . — These  are  called  errors  rather 
than  fallacies  because  they  are  not  due  to  any  process  of 
reasoning,  but  to  mistakes  in  the  aecprisition  of  facts.  They 
give  rise  to  an  error  in  the  conclusion  by  falsifying  the  data 
from  which  the  inference  is  drawn.  This  may  occur  in  two 
ways,  and  against  these  contingencies  the  scientific  student 
should  perpetually  be  on  his  guard.  There  are  no  rules,  how- 
ever, to  determine  when  they  may  occur.  But  the  two 
sources  of  error  in  observation  are  as  follows  : 

1.  Non-Observation. — This  is  simply  the  failure  to  observe 
the  facts  which  determine  the  nature  of  the  inference  or  the 
right  to  make  it.     We  ma}'  fail  to  observe  all  the  facts,  or  we 


364  ELEMENTS  OF  LOGIC 

may  fail  to  observe  those  which  are  essential  to  a  legitimate 
inductive  inference.  The  failure  may  be  due  to  various  causes 
with  which  it  is  hardly  the  business  of  Logic  to  deal  at  length, 
or  farther  than  to  refer  in  general  to  the  admitted  sources  of 
mistake  in  perceptions  ;  but  it  is  an  important  fact  in  scien- 
tific method  that  we  are  liable  to  such  errors  in  the  acquisi- 
tion of  our  data  as  vitiate  the  conclusion,  by  bringing  it  into 
conflict  with  non-observed  facts.  This  non-observation  may 
be  of  some  of  the  cases  or  instances  involved  in,  or  necessary 
to,  the  inductive  inference,  or  of  incidents  in  connection  with 
instances  that  are  observed.  We  may  not  be  able  always  to 
avoid  these  errors,  but  we  should  be  on  our  guard  against 
them. 

2.  Mal-Observation. — Mai  -  Observation  is  not  a  failure  to 
see  facts,  but  is  mistaken  or  distorted  observation.  It  may 
be  either  sensuous  or  intellectual.  Ill  observation  by  the 
senses  may  be  due  to  various  causes,  such  as  defect  of  the 
organs,  indistinctness  of  the  impressions,  indirectness  of  vision 
in  the  case  of  eyesight,  etc.,  so  that  we  may  often  mistake  a 
thing  for  what  it  is  not.  Ill  observation  from  intellectual 
causes  may  be  due  to  mistaking  an  inference  for  a  fact,  to  pre- 
conceived ideas  distorting  the  appearance  of  those  which  are 
facts,  to  the  concentration  of  attention  which  may  prevent  the 
distinct  perception  of  all  that  is  in  the  indirect  field  of  con- 
sciousness, etc.  Some  of  the  last  may  actually  be  a  source  of 
error  to  the  senses  at  the  same  time.  And  there  may  be  cases 
where  there  are  tendencies  in  the  very  mental  and  physical 
organism  to  modify  the  impressions  and  perceptions  which 
consciousness  must  receive.  The  mistaking  of  an  inference 
for  a  fact  of  observation  is  a  very  frequent  error  under  this 
head.  Mr.  Fowler  mentions  the  instance  of  the  objections  at 
first  made  to  the  Copernican  system  of  astronomy.  Peojfle 
claimed  that  the  theory  was  contrary  to  what  they  saw,  when 
the  truth  was  that  what  they  claimed  to  see  was  only  an  infer- 
ence from  certain  facts.  Illusions  often  exhibit  the  same  phe- 
nomenon. We  misinterpret  the  data  of  sense  and  take  the 
result  as  a  fact. 


SCIENTIFIC  METHOD  365 

2d.  Fallacies  of  Inductive  Inference. — It  is  difficult  to 
treat  of  fallacies  in  this  field,  because  there  can  be  no  absolute 
rules  laid  down  to  determine  the  limits  of  correct  inductive  in- 
ference ;  that  is,  to  determine  when  we  may  and  when  we  may 
not  go  beyond  the  data  of  the  premises.  Perhaps  the  only 
rules  applicable  here  are  those  which  determine  the  limits  of 
legitimate  hypothesis.  But  these  have  no  formal  character, 
and  hence  must  be  capable  only  of  material  application  and 
limitations.  In  deductive  reasoning  the  fallacies  are  mostly 
fallacies  of  inference  which  can  be  determined  by  rules.  In 
inductive  reasoning  what  is  called  an  error  of  inference  is 
never,  perhaps,  formal,  but  must  be  material,  and  hence  there 
is  no  way  of  determining  them  until  after  the  process  of  veri- 
fications has  been  applied  and  a  result  established  contrary  to 
our  supposition.  But  if  errors  do  not  arise  from  the  mode  of 
our  inference,  they  may  arise  from  the  assumptions  we  make 
in  the  interpretation  of  phenomena.  Hence  there  may  be 
what  we  may  call  presumptive  fallacies  in  the  inductive  pro- 
cess, which  vitiate  the  conclusiveness  of  the  inference  ;  that  is, 
its  probability  or  certainty,  although  it  may  be  admissible  as 
representing  a  conceivable  case  awaiting  distinct  verification. 
Of  these  fallacies  it  is  sufficient  to  recognize  two  general 
kinds  :  first,  the  mistaking  an  inductive  for  a  deductive  infer- 
ence ;  and  second,  the  mistaking  of  partial  for  total  causes. 

1.  Confusion  of  Inductive  with  Deductive  Inference. — We 
may  often  mistake  the  degree  of  proof  we  have  for  an  asser- 
tion. We  may  have  arrived  at  a  truth  inductively,  and  then 
assume  that  the  process  was  one  of  positive  proof,  or  deductive. 
Or  in  answering  the  demand  for  proof  of  a  proposition,  we 
may  quote  instances  or  facts  which  are  only  the  incidents  from 
which  the  inductive  inference  was  drawn.  They  undoubtedly 
support  the  inference  as  an  inductive  one,  but  if  we  assume 
that  they  prove  it,  we  assume  that  we  have  given  deductive 
reasons,  when  we  have  only  referred  to  facts.  In  deduction 
this  would  be  called  a  petitio  principii.  Thus,  if  I  inferred  that 
the  moon  influenced  the  weather,  or  that  frost  was  caused  by 
cool  calm  nights,  and  supposed  that  my  belief  was  confirmed 


366  ELKMi:.\  TB  OF  LOGIC 

or  proved  by  the  incidents  of  my  observations  and  experience, 
I  would  be  committing  this  fallacy.  It  is  a  very  frequent  er- 
ror, and  in  no  phenomena  is  it  more  frequent  than  in  the  con- 
fusion of  coincidence  or  sequence  with  the  causal  connection. 
This  is  the  non  causa  pro  musa,  or  post  hoc,  t-r<jo  propter  hoc 
fallacy  already  discussed.  We  have  a  right  to  suppose  it  pos- 
sible or  probable,  but  not  as  proved  by  the  circumstances. 
Perhaps,  however,  these  should  not  be  called  inductive  falla- 
cies, so  much  as  fallacies  which  accompany  inductive  reason- 
ing. Their  association  with  it  justifies  the  consideration  of 
them  in  this  connection. 

2.  Confusion  of  Partial  and  Total  Causes. — This  is  merely 
the  presumptive  error  of  supposing  that  what  may  be  one  of 
the  causes  is  the  only  one.  Thus  we  may  infer  from  a  set  of 
observations  that  the  moist  air  of  a  given  region  is  unhealthy, 
when  it  may  be  the  moist  air  in  conjunction  with  the  tempera- 
ture, or  it  may  be  due  to  influences  of  temperature  as  well  as 
moisture.  Or  again,  the  effect  may  be  as  much  determined 
by  the  character  of  the  individual  as  by  the  circumstances  in 
wThich  he  is  placed.  The  error  here,  however,  is  not  in  infer- 
ring a  cause,  but  in  assuming  that  the  discovered  cause  is  the 
only  or  the  whole  cause  of  the  phenomenon,  and  hence  in  the 
application  of  scientific  method  we  need  to  be  as  much  on  our 
guard  against  this  as  against  the  preceding  fallacy. 

This  fallacy  may  take  several  forms,  and  there  are  perhaps 
others  of  an  allied  nature.  But  we  do  not  consider  it  impor- 
tant to  discuss  any  of  them  at  length  in  this  elementary  work. 
The  careful  treatment  of  them  belongs  to  special  treatises  on 
scientific  method.* 

*  For  a  more  complete  discussion  of  this  subject  trie  student  may  con- 
sult the  following  references  :  Jevons  :  Principles  of  Science  ;  Fowler  : 
Inductive  Logic  ;  Mill:  Logic,  Books  III.,  IV.,  and  VI.  ;  Hamilton:  Lec- 
tures on  Logic,  Lects.  XXIV.,  XXV.,  and  XXVI.  ;  Venn:  Empirical 
Logic  Chapters  XIV. -XVIII.,  and  Chapter  XXIV. 


PRACTICAL   QUESTIONS  AND  PROBLEMS. 

Chapter  I. 

1.  What  is  the  distinction  between  "science  "  and  "  art,"  and  how  ap- 
ply it  to  Logic  ? 

2.  Examine  the  merits  of  the  several  definitions  of  Logic. 

3.  Define  the  logical  use  of  the  term  "  thought." 

4.  What  is  Sir  William  Hamilton's  account  of  "  thought  ?  " 

5.  What  is  the  meaning  of  the  term   '"law"  and  such  expressions  as 
"  laws  of  nature  "  and  "  laws  of  thought  '?  " 

6.  Illustrate  the  use  of  the  term  "  law  "  in  the  sciences. 

7.  What  are  the  meanings  of  the  term  "form"  in  common  usage  and 
in  Logic  ?     Also  the  term  "  matter  ?  " 

8.  What  is  the  relation  of  Logic  to  the  other  sciences  ?     Especially  to 
Psychology  ? 

9.  What  are  the  divisions  of  Logic  ?     How  define  the  meaning  of  each 
subdivision  ? 

Chapter  II. 

1.  What  are  the  elements  of  logical  doctrine  ?    From  what  points  of 
view  are  they  to  be  regarded,  and  why  V 

2.  Define  "terms,"  "propositions,"  and  "syllogisms." 

3.  What  are  the  "formal"  elements  of  logical  doctrine,  and  how  de- 
fine them  V 

4.  What  ambiguity  is  found  in  the  word  "  conception  ?  " 

5.  What  is  the  difference  between  a  "  term  "  and  a  "concept  ?  '' 

6.  What  is  the  definition  of  "  concept,"  and  to  what  ambiguities  is  the 
term  exposed  ? 

7.  What  is  a  "  percept,"  and  how  are  concepts  formed  ? 

8.  What  is  meant  by  Perception  or  Apprehension  ? 

9.  What  is  meant  by  "  attribute  "  or  "  individual"  and  "  class  wholes  ?  " 

10.  Explain   the   difference  between  mathematical  and  metaphysical 
or  logical  concepts. 

11.  What  is  the  use  of  Denomination  in  Logic  ? 

12.  Define  Judgment  and  compare  the  process  with  that  of  Conception. 

13.  Define  Reasoning  and  compare  it  with  Conception  ? 


368  ELEMENTS  OF  LOGIC 


Chapter  hi. 

1.  What  are  categorematio  and  syncategorematic  terms  ? 
8.   Define  and  illustrate  Bingular  terms.     When  will  terms  ordinarily 
singular  become  general  ? 
:!.   Define  and  illustrate  general  terms. 

4.  Distinguish  between  distributive  and  collective  terms. 

5.  Examine  the  following  propositions  and  state  the  distributive!  and 
collective  use  of  the  terms  . 

(a)  The  inhabitants  of  Germany  constitute  a  nation. 

(b)  "All  men  find  theix  own  in  all  men's  good, 

And  all  men  join  in  noble  brotherhood." — Tennyson. 

(c)  All  standing  armies  are  dangerous  to  the  .state. 

(d)  Non  omnis  moriar  [i.e.,  I  shall  not  all  die). 

(e)  All  the  men  cannot  lift  this  weight. 
(/)  All  of  the  regiment  was  put  to  flight. 

6.  Define  and  distinguish  concrete  and  abstract  terms. 

7.  Give  illustrations  of  pure  concrete  and  pure  abstract  terms,  and  show 
why  terms  cannot  be  classified  according  to  the  distinction  between  con- 
crete and  abstract 

8.  Indicate  in  the  following  list  the  concrete  and  the  abstract  terms, 
and  those  also,  if  any,  which  may  be  both  : 


Act. 

Ability. 

Plato. 

Production. 

Action. 

Presidency. 

Solitude. 

Warmth. 

Agency. 

Timeliness. 

Dexterity. 

Science. 

Agent. 

Virtue. 

Government. 

Art. 

Beauty. 

Excellence. 

Library. 

Truth. 

Man. 

W  i  sdom. 

Introduction. 

Stone. 

9.  Do  abstract  terms  admit  of  being  plural  ? 

10.  What  disposal  should  be  made  of  general  terms  in  the  classification 
of  concrete  and  abstract  concepts  ? 

11.  What  are  the  common   uses  of  "concrete'"  and  "abstract,"  and 
why  ? 

12.  State  the  views  of  Wundt  on  concrete  and  abstract  terms. 

13.  In  the  following  list  of  terms  show  bow  each  term  may  be  taken  as 
concrete  and  abstract,  according  to  its  meaning  : 

Individuality.  Personality.  Equivocation.  CJovernment. 

Society.  Science.  Philosophy.  Institution. 

14.  Define  and  distinguish  positive,   negative,    and    privative    terms, 
naming  the  marks  of  the  negative. 

15.  Define  and  illustrate  nego-positive  terms,  giving  the  reasons  for 
recognizing  such  a  class. 


PRACTICAL   QUESTIONS  AND  PROBLEMS 


369 


16.  What  is  meant  by  infinitated  conceptions  ? 

17.  How  can  any  term  be  considered  as  the  negative  of  all  others  but 
its  synonym  ? 

18.  Illustrate  absolute  and  relative  terms. 


Chapter  IV. 

1.  What  is  meant  by  the  ambiguity  of  terms  ? 

2.  Defiue  and  illustrate  univocal  terms. 

3.  Distinguish  the  three  kinds  of  equivocal  terms. 

4.  Distinguish  the  three  causes  by  which  the  third  and  most 
class  of  ambiguous  terms  has  been  produced. 

5.  Explain  the  ambiguity  of  the  following  terms,  referring 
cause : 


Chair. 

Man. 

Country. 

Sensation. 

Bill. 

Table. 

Term. 

Letter. 

Commons. 

Mount. 


Paper. 

Stock. 

Air. 

Glass. 

Peer. 

Sense. 

Ball. 

Interest. 

Church. 

Currency. 


Minister. 

Count. 

Period. 

Clerk. 

Order. 

Wood. 

Bull. 

Pole. 

House. 

Fault. 


Volume. 

Scale. 

Feeling. 

Kind. 

State. 

Service. 

Subject. 

Age. 

Virtue. 

Lace. 


important 
each  to  its 

Earth 
Law. 

Art. 

Bolt. 

Star. 

End. 

Class. 

Can. 

Light. 

Dip. 


6.  Explain   and    illustrate    what    is  meant    by  the  generalization   and 
specialization  of  terms,  and  how  they  give  rise  to  ambiguous  ideas. 

7.  How  are  the  laws  of  generalization  and  specialization  applicable  to 
concrete  and  abstract  terms  ? 


Chapter  V. 

1.  What  terms  are  used  to  express  the  same  meaning  as  the  terms  in- 
tension and  extension  ? 

2.  Define  and  distinguish  what  is  meant  by  the  intension  and  extension 
of  concepts. 

3.  Can  abstract  terms  have  both  intension  and  extension  ? 

4.  What  is  the  relation  between  intension  and  extension,  and  how  may 
it  be  formulated  ? 

5.  Between  what  class  of  terms  can  a   comparison   of  extension   be 
made  ? 

6.  When  can  a  comparison  between  them  not  be  drawn,  and  what  is 
peculiar  in  this  respect  regarding  affirmative  and  negative  propositions  ? 

7.  What  can  be  said  about  the  nature  and  accuracy  of  the  law  express- 
ing the  ratio  or  relation  between  intension  and  extension  ? 

24 


370 


ELEMENTS  OF  LOGIC 


8.  How  can  the  relation  be  expressed  by  symbols? 

'.).  What  arguments  can  i>e  produoed  for  tin;  substantial  truth  of  the 

law  ? 

10.  What  is  meant  by  denotative,  connotative,  ami  non-connotative 
terms  F 

11.  Examine  Mill's  dootrine  upou  this  point,  and  Btate  what  modifi- 
oations  air  commended  by  Keynes  and  Fowler? 

12.  Selecl  from  the  Following  li.-t  the  terms  which  belong  to  the  same 
classes,  ami  arrange  them  in  the  order  both  of  the  greatest  intension  and 
the  greatesl  extension : 


Emperor. 
Teacher. 

Baptist. 
Timber. 
Planet. 
Mammalian. 

Frenchman. 


Person. 
Horse. 

Heavenly  body. 
Christian. 

M  iii    i 

Solicitor. 

.Man. 


Animal. 
1  MvM-nter. 
I  QdividuaL 
Jupiter. 
Quadruped. 
II'  in.' 
Word. 


Ruler. 

Organized  substance. 

Lawyer. 

Alexander. 

Napoleon  III. 

Episcopalian. 

<  ireek. 


13.  Select  and  arrange  the  following,  with  the  use  of  triangular  sym- 
bols, so  as  to  show  both  the  order  of  intension  and  extension: 


Animal. 

Machine. 

President. 

Engine. 

Tree. 

Y.i  tcbrate. 

.Manufacture.-. 

Locomotive. 

Gladstone. 

Man. 

Statesman. 

Englishman. 

Substance. 

Element. 

Vegetable. 

Iron. 

Lily. 

MetaL 

Steel. 

Flower. 

Organism. 

Being. 

Lincoln. 

American. 

ClTAPTER    VI. 

1.  "What  are  the  five  predicables  ? 

2.  Define  and  illustrate  the  term  "  property."   What  other  terms  are  its 
equivalents  ? 

3.  What  is  meant  by  essential  property  ?     Illustrate. 

4.  Define  and  illustrate  what  is  meant  by  non-essential  properties  or 
"  accidents." 

5.  What  is  meant  by  universal  and  contingent  accidents  ?     Also  by  the 
term  "  peculiar  "  property  ? 

6.  What  is  meant  by  "  differentia  "  or  difference  ? 

7.  What  is  the  meaning  of  the  term  conferentui,  and  what  is  the  reason 
for  using  it  ? 

8.  Define  "  genus"  and  "species."  illustrating  them  and  showing  how 
they  are  purely  relative  in  their  meaning,  with  two  exceptions. 

9.  What  two  meanings  has  the  term  "  genus  ?  " 

10.  What  is  meant  by  summum  genus  and  infima  species? 


PRACTICAL   QUESTIONS  AND  PROBLEMS  371 


11.  What  is  the  analysis  of  concepts  ? 

12.  What  is  logical  division  ?     Illustrate  and  show  what  is  meant  by 
the  fu  ndamen  t a in  dirisionis. 

13.  Apply  division  as  far  as  possible  to  the  following  list  of  concepts : 

Animal.  Government.  Matter. 

Vegetables.  Man.  Vertebrate. 

Stones.  Book.  School. 

Trees.  Science.  Poetry. 

Races.  Furniture.  Metal. 

14.  What  is  meant  in  division  by  the  terms  super-ordinatc,  subordinate, 
and  co-ordinate  ? 

15.  What  is  dichotomy  and  trichotomy?     Illustrate  the  "tree  of  Por- 
phyry." 

1(3.  Define  and  illustrate  "  partition."     What  is  meant  by  mathematical 
and  logical  partition  ? 

17.  Analyze  the  following  concepts  by  partition  : 


Metal. 

Picture. 

Cathedral. 

Knowledge. 

Ink. 

Iron. 

Stone. 

Religion. 

Money. 

Book. 

Plant. 

House. 

Literature. 

Government. 

Ice. 

German. 

Diamond. 

Virtue. 

Production. 

Wheat. 

18.  Name  and  explain  the  several  kinds  of  definition. 

19.  Compare  the  processes  of  definition  and  division. 

20.  What  are  the  rules  for  correct  definition  ? 

21.  Give  a  logical  definition  of  the  following  terms  : 


Biped. 

Spirit. 

Desk. 

Action. 

Proposition. 

Reptile. 

Water. 

House. 

Book. 

Literature. 

Nation. 

Ability. 

Agent. 

Religion. 

Spectacle. 

Purity. 

Honor. 

Imagination. 

Club. 

Gravitation 

Diet. 

Success. 

Republican. 

Money. 

Race. 

22.  Criticise  the  following  definitions  : 

(a)  A  member  of  the  solar  system  is  anything  over  which  the  sun  has 
continued  influence. 

(b)  A  chair  is  an  object  upon  which  men  sit. 

(c)  An  animal  is  a  being  which  increases  in  size. 

(d)  Death  is  the  opposite  of  life. 

(e)  A  king  is  one  who  exercises  regal  functions. 

(/)  A  gentleman  is  a  man  having  no  visible  means  of  subsistence. 
(g)  Tin  is  a  metal  lighter  than  gold. 
(h)  Government  is  an  association  of  men. 

(i)  Science  is  the  study  of  phenomena  with  a  view  to  a  scientific  knowl- 
edge of  them. 

(j)  Man  is  a  bundle  of  habits. 


372  ELEMENTS  OF  LOGIC 

23.  What  can  be  said  of  the  definition  of  faith  in  the  eleventh  chapter 
of  the  Epistle  to  the  Hebrews  V 

24.  Give  examples  of  indefinable  words  and  explain  why  they  are  inde- 
finable. 

Chapter  VII. 

1.  What  is  the  definition  of  a  judgment  or  proposition  ? 

2.  What  is  meant  by  subject  and  predicate  logically  considered. 

3.  What  are  the  divisions  of  judgments,  and  the  difficulties  suggested 
by  some  of  the  current  divisions  ?     Define  and  illustrate  each. 

4.  How  reduce  disjunctive  propositions  to  hypothetical  ? 

5.  What  is  meant  by  the  quality  of  propositions  ? 

G.   What  is  meant  by  the  division  of  propositions  according  to  quantity  f 

7.  Show  how  the  five  forms  of  propositions  according  to  quantity  can 
be  reduced  to  two. 

8.  Define  and  illustrate  analytic  and  synthetic  judgments. 

9.  Explain  the  difficulty  of   drawing  an   absolute   line   of   distinction 
between  analytic  and  synthetic  judgments. 

10.  What  are  tautologous,  pure,  and  modal  propositions  ? 

11.  What  are  the  three  sources  of  ambiguity  in  propositions  ? 

12.  Define  what  is  meant  by  Inverted  and  Duplex  propositions. 

13.  Illustrate  and  explain  each  of  the  kinds  of  duplex  propositions. 

14.  State  the  complementary  of  the  following  duplex  propositions: 

(a)  None  but  the  wise  can  be  virtuous. 

(b)  All  citizens  except  criminals  and  foreigners  are  not  allowed  to  vote. 

(c)  Only  bipeds  have  hands. 

(fZ)  Man  alone  is  not  obedient  to  his  instincts. 

(e)  Few  persons  are  as  strong  to  resist  temptation  as  they  should  be. 
(/)  Only  those  substances  which  are  not  subject  to  gravity  are  imma- 
terial. 

15.  Explain  the  quantity  of  exclusive  propositions. 

Chapter  VIII. 

1.  Explain  the  meaning  of   "judgments  of    extension"'  and  "judg- 
ments of  intension." 

2.  How  show  the  possibility  of  expressing  a  mathematical  relation  be- 
tween the  subject  and  the  predicate  in  at  least  the  extensive  judgments  ? 

3.  How  show  that  intensive  judgments  have  an  extensive  or  quantita- 
tive, as  well  as  an  intensive  or  qualitative  import? 

4.  What  is  the  difference  between  the  quantity  of  extension  and  quan- 
tity of  intension  expressed  by  propositions  ? 

5.  Symbolize  propositions  A,  E,  I,  and  O,  by  Euler's  diagrams  in  every 


PRACTICAL   QUESTIONS  AND  PROBLEMS         373 

possible  form,  and  reduce  them  to  the  simplest  expression  or  representa- 
tion. 

6.  How  symbolize  the  quantity  of  intension  in  judgments  ? 

7.  Define  what  is  meant  by  the  distribution  of  the  subject  and  predi- 
cate, and  give  the  rules  for  it. 


Chapter  IX. 

1.  Explain  the  meaning  of  opposition. 

2.  What  is  meant  by  the  terms  contrary,  contradictory,  subalterns, 
subalternans,  subalternate,  and  sub-contrary  ? 

3.  If  we  assume  any  one  of  the  four  propositions,  A,  E,  I,  and  O,  to  be 
false,  what  follows  in  regard  to  the  others  ? 

4.  How  must  we  express  the  opposite  of  any  singular  proposition  ? 

5.  How  treat  the  relation  between  the  two  propositions,  "Daniel  Web- 
ster was  an  American,"  and  "  Daniel  Webster  was  not  an  Englishman  ?" 

6.  How  are  propositions  best  proved  or  disproved,  and  what  difficulty 
do  universal  propositions  present  in  disputation  ? 

7.  Select  pairs  of  the  following  propositions  and  arrange  them  so  as  to 
show  all  the  various  relations  of  opposition  illustrated  by  them : 

(a)  All  men  are  mortal. 

(b)  Some  men  are  not  mortal. 

(c)  No  men  are  mortal. 

(d)  Some  men  are  not  mortal. 

(e)  Most  men  are  mortal. 
(/)  All  men  are  not  mortal. 
(g)  Not  all  men  are  mortal. 
(h)  Only  men  are  mortal, 
(i)  Few  men  are  not  mortal. 
(j)  Only  mortals  are  men. 

{k)  All  men  except  a  few  are  mortal. 

8.  Examine  the  relation  between  the  following  propositions  and  exam- 
ples of  assertion,  and  state  what  is  implied  by  a  given  assertion  against 
an  opponent. 

(a)  One  man  asserts  that  all  men  are  wise,  and  another  that  they  are  all 
ignorant. 

(b)  Apples  are  a  species  of  fruit,  but  they  are  not  an  ordinary  vege- 
table. 

(c)  Free-trade  lowers  prices  and  protection  raises  them. 

(d)  The  leaders  of  one  party  assert  that  Mr.  A  will  be  elected  presi- 
dent, and  the  leaders  of  the  other  that  Mr.  B  will  be  elected  president. 

(e)  Mr.  X  asserts  that  not  a  nail  was  made  in  this  country  before  1801. 


374  ELEMENTS  OF  LOGIC 

So  far  is  this  statement  of  X  from  being  true  that  in  185G  there  were 
2,645  nail  machines  in  operation  in  this  country  with  an  output  of  86,462 
tons,  and  in  185!)  as  many  as  4,686,207  pounds  of  nails  were  exported. 

(/)  "  Will  the  educated  women  marry  ?  So  queried  one  of  our  alum- 
nae in  a  recent  magazine  article.  The  review  roll  of  our  alumna-  shows 
that  of  7<i  Ladies  who  graduated  in  our  classes,  >V2  have  already  married." 

(g)  "  Great  efforts  are  made  to  show  that  a  general  glut  of  the  market 
is  not  a  source  of  evil,  and  that  the  appreciation  of  gold  has  been  the 
cause  of  the  extraordinary  fall  in  prices.  To  those  who  hold  this  view  I 
would  put  the  question  why  ivory  and  whalebone  have  not  fallen  in 
price,  but,  on  the  contrary,  have  steadily  risen  in  price  during  the  last 
decade." — Cotter  Morrison,  Servia  of  Man,  Preface,^,  x.-xi. 

(7i)  "  An  import  duty  is  not  a  tax,  and  yet  raises  the  price  of  articles  to 
the  consumer. 

"  The  object  of  raising  prices  to  the  consumer  is  to  enable  the  domestic 
manufacturer  to  pay  higher  wages  to  his  workmen. 

"The  duty  does  not  raise  prices  to  the  consumer,  but,  on  the  contrary, 
lowers  them. 

"  The  manufacturers  are  enabled  to  pay  higher  wages  in  spite  of  this 
fall  in  prices." 

Chapter  X. 

1.  What  is  the  meaning  of  inference?  Of  immediate  inference?  Of 
mediate  inference  ? 

2.  Define  and  illustrate  Conversion. 

3.  What  are  the  rules  for  Conversion,  and  what  is  meant  by  the  con- 
vertend  and  the  converse  ? 

4.  How  many  kinds  of  Conversion  are  there  ?  How  are  they  variously 
applied  to  propositions  A,  E,  I,  and  O  ? 

5.  What  is  true  of  the  conversion  of  definitions,  and  of  singular  propo- 
sitions with  a  singular  predicate,  and  why  V 

6.  What  is  Conversion  by  Negation,  to  what  proposition  is  it  applied, 
and  how  '?     Show  whether  it  is  a  legitimate  form  of  conversion  or  not. 

7.  Explain  and  illustrate  the  process  of  Obversion,  and  apply  it  to  the 
four  propositions. 

8.  Explain  and  illustrate  Contraversion  or  Contraposition,  and  show 
why  it  cannot  be  applied  to  proposition  I. 

9.  Explain  in  what  sense  Contraversion  is  a  process  of  immediate  infer- 
ence, if  it  be  so  at  all. 

10.  Define  and  illustrate  the  process  of  Inversion,  and  show  to  what 
propositions  it  is  applicable,  and  to  what  propositions  it  is  not  applicable. 

11.  What  is  meant  by  Inference  by  Contribution,  and  what  are  its  di- 
visions ?     Illustrate  each. 


PRACTICAL   QUESTIONS  AND  PROBLEMS  375 

12.  When  will  Inference  by  Added  Determinants  and  by  Complex  Con- 
ception be  invalid  ? 

13.  State  the  logical  process  by  which  we  pass  from  each  of  the  follow- 
ing propositions  to  the  succeeding  one  : 

(a)  All  metals  are  elements. 

(b)  No  metals  are  non-elements. 

(c)  No  non-elements  are  metals. 

(d)  All  non-elements  are  not  metals. 

(e)  All  metals  are  elements. 
(/)  Some  elements  are  metals. 
(g)  Some  metals  are  elements. 
(h)  No  metals  are  elements. 

14.  Convert  the  following  propositions : 
(1.)  Every  man  is  a  biped. 

(2.)  No  triangle  has  one  side  equal  to  the  sum  of  the  other  two. 

(3.)  Some  books  are  dictionaries. 

(4.)  "  Every  consciousness  of  relation  is  not  cognition." 

(5.)  Vegetables  only  are  deciduous. 

(6.)  A  stitch  in  time  saves  nine. 

(7.)  Perfect  happiness  is  impossible. 

(8.)  Few  are  acquainted  with  themselves. 

(9.)  No  one  is  free  who  does  not  control  himself. 

(10.)  Good  orators  are  not  always  good  statesmen. 

(11.)  Some  inorganic  substances  do  not  contain  carbon. 

(12.)  All  men  are  not  born  equal. 

(13.)  Only  the  brave  deserve  the  fair. 

(14.)  No  one  is  a  hero  to  his  valet. 

(15.)  He  jests  at  scars  who  never  felt  a  wound. 

(16.)   Uneasy  lies  the  head  that  wears  a  crown. 

(17.)  Better  late  than  never. 

(18.)  A  certain  man  had  a  fig  tree. 

(19.)  Familiarity  breeds  contempt. 

(20. )  Every  mistake  is  not  culpable. 

(21.)  I  shall  not  all  die  (non  omnis  moriar). 

(22. )  Not  many  of  the  metals  are  brittle. 

(23.)  Great  is  Diana  of  the  Ephesians. 

(24.)  Talents  are  often  misused. 

(25.)  Romulus  and  Remus  were  twins. 

(26.)  Some  books  are  to  be  read  only  in  part. 

(27.)  Nothing  is  praiseworthy  but  virtue. 

(28.)  Two  blacks  will  not  make  a  white. 

(29.)  Not  one  of  the  Greeks  at  Thermopylae  escaped. 

(30.)  No  one  is  always  happy. 


370  ELEMENTS  OF  LOGIC 

(31.)  Metals  are  all  good  conductors  of  heat. 

(32.)  There  is  none  good  but  one. 

(33.)  All  that  glitters  is  not  gold. 

(34.)  He  can't  be  wrong  whose  life  is  in  the  right. 

15.  State  the  relation  between  the  following  propositions  as  indicated 
by  the  figures  in  parentheses  at  the  end  of  each  proposition : 

(1.)  Good  men  are  wise. 
(2.)  Unwise  men  are  not  good  (1.). 
:{.)  Some  wise  men  are  good  (1.). 
(4.)  No  good  men  are  unwise  (1.)  (2.). 
(5.)  Some  unwise  men  are  not  good  (2.)  (4.). 
(6.)  Some  good  men  are  wise  (1.)  (3.). 
(7.)  No  good  men  are  wise  (1.)  (4.)  (6.)  (3.). 
(8.)  Some  good  men  are  not  wise  (1.)  (3.)  (6.)  (7.). 
(9.)  No  unwise  men  are  good  (1.)  (2.)  (4.)  (5.)  (8.). 
(10.)  No  wise  men  are  good  (1.)  2.)  (6.)  (7.)  (8.). 

16.  What  is  the  logical  relation,  if  any,  between  the  two  following  prop- 
ositions :  "A  false  balance  is  an  abomination  to  the  Lord,  but  a  just 
weight  is  his  delight." 

17.  What  can  be  inferred  by  Obversion,  Conversion,  and  Contraversion 
from  the  following  proposition  :  "The  angles  at  the  base  of  an  isosceles 
triangle  are  equal." 

18.  State  the  Contraverse  and  Obverse  of  propositions  (1.)  (5.)  (6.)  (7.) 
(9.)  (1.5.)  (19.)  (23.)  (31.)  and  (33.)  under  question  14. 

19.  Can  we  logically  infer  that  because  heat  expands  bodies,  therefore 
cold  contracts  them  ? 

20.  State  the  relation  between  the  following  three  propositions:  "  The 
voluntary  muscles  are  all  striped,  and  the  unstriped  are  all  involuntary, 
but  a  few  of  the  involuntary  muscles  are  striped." 


Chapter  XI. 

(1.)  What  is  meant  by  a  middle  term  ?  Explain  what  is  meant  by  ma- 
jor and  minor  terms,  and  show  how  each  of  the  three  terms  can  be  dis- 
tinguished in  a  syllogism. 

(2.)  State  the  rules  of  the  syllogism 

(3.)  Define  what  is  meant  by  a  formal  fallacy,  and  state  what  three 
kinds  of  them  occur. 

(4.)  How  can  we  symbolize  by  diagrams  the  several  kinds  of  formal 
fallacy. 

(5.)  In  the  following  syllogisms  and  reasonings  point  out  the  major, 


PRACTICAL   QUESTIONS  AND  PROBLEMS  377 

middle,  and  minor  terms,  with  the  corresponding  premises  and  conclu- 
sion : 

(a)  All  men  are  fallible.  (b)  Platinum  is  a  metal. 

All  kings  are  men.  All  metals  are  heavy. 

.\  All  kings  are  fallible-  .*.  Platinum  is  heavy. 

(c)  Cattle  are  ruminants.  ((/)  Iron  is  a  metal. 

Horses  are  not  ruminants.  Metals  are  substances. 

.  \  Cattle  are  not  horses.  .  \  Some  substances  are  iron. 

(e)  He  is  wise,  because  he  knows  what  his  interest  is  and  whoever 
knows  his  interest  is  wise. 

Chapter  XII. 

1.  Define  what  is  meant  by  the  Mood  of  a  syllogism.  Illustrate  and 
show  how  many  combinations  are  possible  with  the  four  propositions  A, 
E,  I,  O. 

2.  Mark  those  moods  which  are  invalid  and  state  the  reason. 

3.  What  kind  of  a  conclusion  can  be  drawn  from  the  following  prem- 
ises, AA,  EA,  IA,  AE.  OA,  EI. 

4.  Define  what  is  meant  by  the  Figure  of  a  syllogism,  and  illustrate 
each  Figure. 

5.  What  is  meant  by  a  weakened  conclusion  ? 

6.  What  peculiar  value  attaches  to  each  of  the  first  three  Figures  of 
syllogism  ? 

7.  Show  in  what  figures  the  following  premises  give  valid  conclusions, 
AA,  AE,  AI.  EA,  OA,  EI,  AO. 

8.  Show  that  O  cannot  stand  as  a  premise  in  the  first  Figure,  as  major 
premise  in  the  second  Figure,  and  as  minor  premise  in  the  third  Figure. 

9  What  fallacies  would  be  committed  by  having  A  as  a  conclusion  in 
any  Figure  but  the  first  ? 

10.  Why  can  we  prove  only  negative  propositions  in  the  second  Fig- 
ure ? 

11  What  fallacy  is  committed  if  the  minor  premise  of  the  first  Figure 
be  negative  ? 

12.  If  one  premise  be  O,  what  must  the  other  be  ? 

13.  What  are  the  Moods  and  Figures  of  the  following  syllogisms  ?  Name 
the  valid  and  the  invalid  Moods. 

(a)  Some  M's  are  P's  (b)  All  P's  are  M's. 

No  S's  are  M's.  No  M's  are  S's. 

. \  Some  P's  are  not  S's.  .  *.  No  P's  are  S's. 

(c)  All  S's  are  M's.  (d)  No  M's  are  P's. 

No  P's  are  M's.  All  M's  are  S's 

.-.  Some  S's  are  not  P's.  .*.  Some  S's  are  not  P's. 


378  ELEMENTS  OF  LOGIC 

(e)  All  feathered  animals  are  vertebrates. 
No  reptiles  are  feathered  animals. 

.•.  Some  reptiles  are  not  vertebrates. 

(f)  All  vices  an-  reprehensible. 
Emulation  is  not  reprehensible. 

.'.  Emulation  is  not  a  vice. 

(g)  All  men  arc  rational  beings. 

All  Caucasians  are  rational  beings 
. '.  All  Caucasians  are  men. 
(h)  All  vices  are  reprehensible. 

Emulation  is  not  a  vice. 
.  •.  Emulation  is  not  reprehensible. 
(i)  Only  citizens  are  voters. 

A,  B,  C  are  voters. 
. -.  A,  B,  C  are  citizens. 

14.  Deduce  conclusions,  stating  Moods  and  Figures,  from  the  following 
premises  : 

(<i)  All  planets  are  heavenly  bodies. 
No  planets  are  self-luminous. 

(b)  All  Europeans  are  Caucasians. 
All  Caucasians  are  white. 

(c)  All  lions  are  carnivorous  animals. 

No  carnivorous  animals  are  devoid  of  claws. 
(cl)  Some  animals  are  quadrupeds. 

All  quadrupeds  are  vertebrates. 
(e)  Oak-trees  are  not  evergreen. 

Pine-trees  are  evergreen. 

15.  Invent  examples  showing  true  conclusions  with  false  premises. 

Chatter  XIII. 

1.  Explain  the  mnemonic  lines  which  represent  the  valid  Moods  and 
Figures,  and  the  reduction  of  the  last  three  Figures  to  the  first. 

2.  Construct  syllogisms  in  Camenes,  Cesare,  Ferison,  Fesapo,  Camestres, 
and  Datisi.  and  reduce  them  to  the  corresponding  Moods  in  the  first  Figure. 

3.  What  is  the  difference  between  Direct  and  Indirect  Reduction. 

4.  Why  cannot  Baroko  and  Bokardo  be  reduced  directly  ?     Show  how 
they  may  be  reduced  indirectly. 

5.  Apply  indirect  reduction  to  Cesare  and  Camenes. 

Chapter  XIV. 

1.  Define  Prosyllogism,  Episyllogism,  Enthymeme,  Epicheirema,  and 
Sorites.     Illustrate  each. 


PRACTICAL   QUESTIONS  AND  PROBLEMS  379 

2.  What  is  the  difference  between  the  regressive  and  the  progressive 
Sorites  ? 

3.  Upon  what  is  the  distinction  between  the  three  orders  of  Enthymeme 
based  ? 

4.  Produce  an  example  of  syllogism  in  which  there  are  two  Prosyllo- 
gisms. 

5.  Complete  the  following  syllogisms  : 

(it)  Europeans  are  Caucasians  because  they  are  white. 

(b)  Since  he  was  directed  to  deliver  the  message,  and  did  not,  I  am  at 
liberty  to  do  as  I  please. 

(c)  We  cannot  know  what  is  false  because  knowledge  cannot  be  decep- 
tive. 

{d)  A  is  B,  because  C  is  B 
E  is  A,  because  G  is  E 
.'.  EisB. 
(e)  A  manor  cannot  begin  at  this  day,  because  a  court-baron  cannot  now 
be  founded. 

Chapter  XV. 

1.  What  is  hypothetical  reasoning  ?  Illustrate.  State  the  signs  of  it  and 
define  what  is  meant  by  the  antecedent  and  the  consequent. 

2.  What  are  the  valid  and  the  invalid  forms  of  hypothetical  reasoning  ? 
What  is  meant  by  modus  ponens  and  modus  tollens  ? 

3.  How  can  hypothetical  syllogisms  be  reduced  to  the  categorical  ? 

4.  What  moods  and  figures  of  the  categorical  syllogism  do  the  modus 
ponens  and  modus  tollens  of  the  hypothetical  syllogism  belong  ? 

5.  What  formal  fallacies  are  committed  by  the  invalid  forms  of  hypo- 
thetical syllogism  ?     Prove  by  reduction  to  the  categorical. 

6.  Examine  the  following  instances  of  hypothetical  reasoning : 

(a)  Rain  has  fallen,  if  the  ground  is  wet  ;  but  the  ground  is  not  wet ; 
therefore  rain  has  not  fallen. 

(b)  If  rain  has  fallen,  the  ground  is  wet ;  but  rain  has  not  fallen  ;  there- 
fore the  ground  is  not  wet. 

(c)  The  ground  is  wet  if  rain  has  fallen  ;  the  ground  is  wet  ;  therefore 
rain  has  fallen. 

(d)  If  the  ground  is  wet,  rain  has  fallen ;  but  rain  has  fallen  ;  therefore 
the  ground  is  wet. 

(e)  If  a  man  cannot  make  progress  toward  perfection,  he  must  be  a 
brute  ;  but  no  man  is  a  brute  ;  therefore  every  man  is  capable  of  such 
progress. 

(/)  If  two  and  two  may  make  five  in  some  other  planet,  Mill's  opinion 
about  the  matter  is  correct  ;  but  they  do  not  make  five  in  any  place  and 
hence  Mill  is  wrong. 


3S0  ELEMENTS  OF  LOGIC 


Chapter  XVI. 

1.  Define  and  illustrate  Disjunctive  Reasoning. 

2.  Upon  what  does  incomplete  disjunction  depend  ? 

3.  Classify  the  forms  of  disjunctive  syllogism,  and  show  how  they  may 
be  reduced  to  either  the  hypothetical  or  the  categorical  form. 

4.  To  what  fallacy  is  disjunctive  reasoning  incident  ? 

5.  Examine  the  following  cases  of  disjunctive  reasoning  : 

(a)  Criminals  are  either  good  or  bad. 
They  are  bad. 

.  •.  They  are  not  good. 

(b)  The  weather  will  be  either  clear  or  warm. 
It  will  not  be  warm. 

.  \  Therefore  it  will  be  clear. 

(c)  Aristotle  was  either  very  talented  or  very  industrious. 
He  was  very  industrious. 

.*.  He  was  not  very  talented. 

(d)  If  the  government  enacts  such  a  law  it  must  either  adopt  socialism 
or  go  into  bankruptcy.  But  it  will  not  enact  such  a  law,  and  hence  there 
is  no  danger  of  either  socialism  or  bankruptcy. 

(e)  If  pain  is  severe  it  will  be  brief,  and  if  it  last  long  it  will  be  slight; 
it  is  either  severe  or  it  lasts  long,  and  therefore  will  be  either  brief  or 
slight. 

(/)  If  capital  punishment  involves  cruelty  to  its  victims  it  ought  to 
be  abolished  in  favor  of  some  other  penalty  ;  if  it  does  no  good  for  society 
it  should  also  be  abolished.  But  it  either  involves  cruelty  to  its  victims 
or  does  no  good  to  society,  and  hence  it  ought  to  be  abolished. 

Chapter  XVII. 

1.  Define  the  term  "fallacy  "  and  explain  what  is  meant  hy  formal  and 
material  fallacies. 

2.  Explain  what  is  meant  by  the  fallacies  of  Amphibology  and  of  Ac- 
cent. 

3.  Give  outline  form  of  the  classification  of  fallacies. 

4.  Illustrate  the  formal  fallacies  and  also  those  of  Amphibology  and 
Accent. 

CHAPTEB    XVIII. 

1.  What  are  the  grounds  upon  which  a  twofold  division  of  material 
fallacies  may  rest  ? 

2.  Explain  what  is  meant  by  the  following  terms  :  petitio  principii,  non 


PRACTICAL   QUESTIONS  AND  PROBLEMS         381 

sequitur,  non  causa  pro  causa,  or  post  hoc,  ergo  propter  hoc,  circulus  inpro- 
bando,  assumptio  non  probata,  and  iijnoratio  elenchi. 

3.   Explain  the  fallacies  of  Quantity  and  Quality,  or  those  of  Composi- 
tion and  Division  and  of  Accident. 


Chapter  XIX. 

1.  What  is  meant  by  the  quantification  of  the  predicate? 

2.  What  effect  upon  the  number  of  propositions  to  be  considered  by 
Logic  is  produced  by  quantifying  the  predicate  explicitly  ? 

3.  How  does  the  quantification  of  the  predicate  effect  the  process  of 
conversion  ? 

4.  What  additional  rule  must  be  added  to  the  rules  of  the  syllogism  if 
we  accept  the  doctrine  of  the  quantification  of  the  predicate  ? 

5.  What  is  peculiar  about  definitions  and  exclusive  propositions  in  re- 
lation to  this  doctrine  ? 

Chapter  XX. 

1.  What  is  the  nature  of  mathematical  propositions  ? 

2.  What  effect  do  they  produce  upon  the  Figures  of  the  syllogism  ? 

3.  Write  out  the  list  of  valid  Moods  when  propositions  are  mathemati- 
cal, and  show  why  each  one  is  valid  that  is  not  valid  in  ordinary  reason- 
ing. 

4.  How  does  mathematical  reasoning  simplify  the  symbolic  representa- 
tion of  the  syllogism  V 

5.  What  characteristic  of  conceptions  appears  in  mathematical  reason- 
ing, and  what  is  excluded  ? 

G.  What  is  meant  by  Traduction,  or  traductive  reasoning  ? 

7.  How  can  you  treat  syllogisms  which  have  been  called  irregular  by 
Jevons  and  others  V  Invent  instances  and  illustrate.  How  are  they  re- 
lated to  the  principles  of  mathematical  reasoning  ? 

Chapter  XXI. 

1.  What  is  the  general  nature  of  the  "Laws  of  Thought"  and  what 
are  their  divisions  ? 

2.  Define  and  illustrate  the  laws  of  Identity,  Contradiction,  Excluded 
Middle,  and  Sufficient  Reason. 

3.  Enumerate  the  secondary  laws  of  thought. 

4.  How  are  the  primary  laws  related  to  the  principles  enunciated  in 
Formal  Logic  r 


382  ELEMENTS  OF  LOGIC 


Chapter  XXII. 

1.  What  views  have  been  taken  regarding  the  nature  of  inductive  rea- 
soning ? 

2.  What  two  recognized  forms  of  induction  are  to  be  considered  in  the 
definition,  and  how  are  they  to  be  treated  ? 

3.  What  is  the  difference  between  generalization  by  enumeration  and 
inductive  inference  ? 

4.  What  is  the  essential  conception  involved  in  the  idea  of  induction 
since  the  time  of  Bacon,  and  how  did  it  arise  ? 

5.  Illustrate  Inductive  reasoning  and  compare  it  with  the  deductive. 

6.  State  the  form  of  the  inductive  syllogism,  illustrate  and  compare  it 
with  the  deductive. 

7.  What  are  the  reasons  for  regarding  inductive  reasoning  as  purely 
qualitative,  and  deductive  as  mainly,  if  not  wholly,  quantitative? 

8.  What  is  meant  by  the  division  of  inductive  inferences  into  statical 
and  dynamical  ' 

9.  What  are  the  so-called  "principles"  of  induction?  Define  their 
meaning  and  relation  to  inductive  reasoning. 

10.  What  is  meant  by  the  principles  or  canons  of  Agreement  and  Dif- 
ference, and  their  functions  in  inductive  reasoning  ? 

Chapter  XXIII. 

1.  Explain  the  meaning  of  scientific  method,  also  what  is  meant  by  the 
methods  of  discovery  and  of  instruction. 

2.  Define  and  illustrate  Deductive  Method.  State  its  divisions  and  ex- 
plain their  relation  to  the  method. 

8.  Explain  what  is  meant  by  the  Inductive  Method  as  compared  with 
inductive  reasoning. 

4.  Define  and  illustrate  the  primary  processes  of  ascertaining  new 
knowledge. 

5.  What  are  the  rules  to  be  observed  in  forming  hypotheses  ? 

6.  Explain  the  process  of  verification,  stating  its  relation  to  inductive 
inference. 

7.  What  are  the  two  or  more  kinds  of  verification  ?  Explain  those 
which  seem  also  to  be  conditions  or  factors  of  the  inductive  inference, 
namely,  Observation  and  Experiment. 

8.  What  are  the  Inductive  Methods,  and  how  do  they  serve  as  verify- 
ing processes. 

9.  Show  how  deduction  may  enter  as  a  process  of  verification. 

10.  Define  and  illustrate  what  are  called  the  fallacies  of  induction. 
Why  distinguish  between  errors  of  observation  and  errors  of  inference  ? 


PEACTICAL  EXERCISES 

DEDUCTIVE 

The  student  is  expected  to  examine  the  following  arguments  ;  to  state 
the  mood  and  figure  of  the  syllogism  where  necessary  ;  to  complete  im- 
perfect syllogisms  ;  to  indicate  the  instances  of  valid  and  invalid  reason- 
ing ;  and  if  invalid,  to  state  whether  the  fallacy  is  formal  or  material, 
and  what  the  particular  fallacy  is.  He  should  also  be  prepared  to  make 
any  resolution  of  the  propositions  and  syllogisms  which  the  rules  of  Logic 
would  enable  him  to  do. 

1.  None  but  animals  are  quadrupeds. 
Horses  are  quadrupeds. 
Therefore  horses  are  animals. 

2.  Personal  deformity  is  an  affliction  of  nature. 
Disgrace  is  not  an  affliction  of  nature. 
Therefore  personal  deformity  is  not  a  disgrace. 

3.  All  roses  are  beautiful. 
Lilies  are  not  roses. 
Therefore  lilies  are  not  beautiful. 

4.  All  paper  is  useful  ;  and  all  that  is  useful  is  a  source  of  comfort  to 
men  ;  therefore  all  paper  is  a  source  of  comfort  to  men. 

5.  Some  statesmen  are  also  authors  ;  for  such  are  Burke,  Macaulay. 
Gladstone,  Lord  Russell,  etc. 

6.  Some  philosophers  are  logicians. 

No  logicians  are  ignorant  of  the  works  of  Aristotle. 
Therefore  some  philosophers  are  not  ignorant  of   the  works  of  Aris- 
totle. 

7.  No  persons  destitute  of  imagination  are  true  poets. 
Some  persons  destitute  of  imagination  are  good  logicians. 
Therefore  some  true  poets  are  not  good  logicians. 

8.  This  explosion  must  have  been  occasioned  by  gunpowder  ;  for  noth- 
ing else  would  have  possessed  sufficient  force. 

9.  If  Cassar  was  a  tyrant,  he  deserved  to  die. 
Caesar  was  not  a  tyrant. 

Therefore  he  did  not  deserve  to  die. 

10.  Good  is  the  object  of  moral  approbation.  The  highest  good  is, 
therefore,  the  ultimate  object  of  such  approbation,  the  end  of  action. 


384  ELEMENTS  OF  LOGIC 

11.  Every  one  desires  his  own  good. 

Justice  and  temperance  are  every  one's  good. 
Therefore  every  one  desires  to  be  just  and  temperate. 

12.  "  But  it  is  doubtful  yet  whether  C.'esar  will  come  forth  to-day  or  not. 
For  he  is  superstitious  grown  of  late.'' 

13.  Every  man  should  be  moderate  ;  for  excess  will  cause  disease. 

14.  All  Parisians  are  Frenchmen. 
No  Chinese  are  Parisians. 

Therefore  some  Chinese  are  not  Frenchmen. 

15.  Some  men  are  not  virtuous. 
All  Americans  are  men. 

Some  Americans  are  not  virtuous. 

16.  Blessed  are  the  merciful ;  for  they  shall  obtain  mercy. 

17.  As  almost  all  the  organs  of  the  body  have  a  known  use,  the  spleen 
must  have  some  use. 

18.  Some  of  the  inhabitants  of  the  globe  are  more  civilized  than  others. 
No  savages  are  more  civilized  than  other  races. 

Some  savages  are  not  inhabitants  of  the  globe. 

19.  Cogito,  ergo  sum  (I  think,  therefore  I  am). 

20.  He  must  be  a  Mohammedan,  for  all  Mohammedans  hold  these  opin- 
ions. 

21.  He  must  be  a  Christian,  for  only  Christians  hold  these  opinions. 

22.  Logic  is  either  a  science  or  an  art. 
It  is  a  science. 

Therefore  it  is  not  an  art. 

23.  No  idle  person  can  be  a  successful  writer  of  history  ;  therefore, 
Hume,  Macaulay,  Hallam,  and  Grote  must  have  been  industrious. 

24.  Who  spareth  the  rod  hateth  his  child  ;  the  parent  who  loveth  his 
child  therefore  spareth  not  the  rod. 

25.  The  coronation  took  place  either  at  Paris,  Berlin,  or  Vienna  ;  it  did 
not  occur  at  Paris  or  Berlin,  and  consequently  must  have  occurred  at 
Vienna. 

26.  Every  moral  man  obeys  the  law  ;  every  citizen  does  not  do  so,  and 
therefore  is  not  moral. 

27.  Rational  beings  are  accountable  for  their  conduct  ;  brutes,  not  be- 
ing rational,  are  therefore  free  from  responsibility. 

28.  All  valid  syllogisms  have  three  terms. 
This  syllogism  has  three  terms. 

This  syllogism  is  therefore  valid. 

29.  All  syllogisms  are  valid  that  have  three  terms. 
This  syllogism  has  three  terms. 

Therefore  this  syllogism  is  valid. 

30.  Comets  are  heavy  matter  ;  for  otherwise  they  would  not  obey  the 
law  of  gravitation. 


PRACTICAL  EXERCISES  385 

31.  A  charitable  man  has  no  merit  in  relieving  distress,  because  he 
merely  does  what  is  pleasing  to  himself. 

32.  None  but  savages  were  in  America  when  it  was  discovered. 
The  Hottentots  were  savages. 

Therefore  they  were  in  America  when  it  was  discovered. 

33.  None  but  despots  possess  absolute  power. 
The  Czar  of  Russia  is  a  despot. 
Therefore  he  possesses  absolute  power. 

34.  Bacon  was  a  great  philosopher  and  statesman,  and  as  he  was  also 
a  lawyer  we  may  infer  that  any  lawyer  may  be  a  great  philosopher  and 
statesman. 

35.  Mathematical  studies  undoubtedly  improve  the  reasoning  powers  ; 
but  as  Logic  is  not  a  mathematical  study  we  may  conclude  that  it  does 
not  improve  our  reasoning  powers. 

36.  If  a  man  cannot  obey  the  law  he  must  be  either  a  mere  machine  or 
a  demon  ;  but  no  man  is  either  of  these,  and  hence  he  must  be  able  to 
obey  the  law. 

37.  Whatever  tends  to  withdraw  the  mind  from  pursuits  of  a  low  nat- 
ure deserves  to  be  promoted  ;  classical  learning  does  this,  since  it  gives 
us  a  taste  for  intellectual  enjoyments ;  therefore  it  deserves  to  be  pro- 
moted. 

38.  Alexander  the  Great  was  the  son  of  King  Philip,  and  therefore 
King  Philip  was  the  father  of  Alexander  the  Great. 

39.  He  that  withholdeth  corn,  the  people  shall  curse  him.  But  bless- 
ing shall  be  upon  the  head  of  him  that  selleth  it. — Proverbs  of  Solomon. 

40.  If  virtue  is  involuntary,  vice  is  involuntary. 
Vice  is  voluntary. 

Therefore  virtue  is  voluntary. 

41.  All  civilized  people  are  inhabitants  of  the  earth.  Few  Indians  are 
civilized,  and  therefore  few  Indians  are  inhabitants  of  the  earth. 

42.  To  improve  is  to  change,  and  to  be  perfect  is  to  have  changed  often. 
What  hope  can  we  entertain  of  those  who  oppose  change  ? 


43.  The  Germans    are  a  nation.     Bismarck,   Stein,   Kant,    and  Hege 
were  Germans,  and  hence  must  have  been  a  nation. 

44.  The  right  should  be  enforced  by  law.     Hence  as  the  exercise  of  the 
suffrage  is  a  right,  it  should  be  enforced  by  law. 

45.  Napoleon  was  not  a  great  emperor  ;  for  though  he  would  have  been 
great  had  he  succeeded  in  retaining  his  power,  he  did  not  do  so. 

46.  Nothing  is  better  than  wisdom  ;   dry  bread  is  better  than  nothing  ; 
therefore  dry  bread  is  better  than  wisdom. 

47.  Knowledge  is  of  no  use  to  any  one  in   preventing  him  from   com- 
mitting crimes  ;  for  we  hear  every  day  of  frauds  and  forgeries   which 

25 


386  ELEMENTS  OF  LOGIC 

never  would  have  been  committed  had  the  person  not  learned  to  read  and 
write. 

48.  The  end  of  punishment  is  either  the  protection  of  society  or  the 
reformation  of  the  individual.  Capital  punishment  ought,  therefore,  to 
be  abolished,  because  it  neither  prevents  crimes  of  violence,  nor  protects 
society,  nor  does  it  reform  the  individual. 

49.  Wealth  is  value  ;  value  is  purchasing  power  ;  purchasing  power  is 
the  product  of  labor,  and  the  product  of  labor  is  property  ;  therefore 
wealth  is  property. 

50.  Every  rule  has  exceptions  ;  this  is  a  rule,  and  therefore  has  excep- 
tions ;  therefore  there  are  some  rules  that  have  no  exceptions. 

51.  All  who  think  this  man  innocent  think  he  should  not  be  punished  ; 
you  think  he  should  not  be  punished  ;  therefore  you  think  him  innocent. 

52.  All  who  think  this  man  innocent  think  he  should  not  be  punished  ; 
you  think  he  should  be  punished  ;  therefore  you  do  not  think  him  inno- 
cent. 

53.  Haste  makes  waste,  and  waste  makes  want.  A  man,  therefore, 
never  loses  by  delay. 

54.  All  equilateral  triangles  are  equiangular,  and  therefore  all  equian- 
gular triangles  are  equilateral. 

55.  Only  the  virtuous  are  truly  noble  ;  some  who  are  called  noble  are 
not  virtuous ;  therefore  some  who  are  called  noble  are  not  truly  noble. 

56.  For  those  who  are  bent  on  cultivating  their  minds  by  diligent  study 
the  incitement  of  academic  honors  is  unnecessary ;  and  it  is  ineffectual 
for  the  idle  and  such  as  are  indifferent  to  mental  improvement ;  therefore 
the  incitement  of  academic  honors  is  either  unnecessary  or  ineffectual. 

57.  Logic  as  it  was  cultivated  by  the  schoolmen  proved  a  fruitless  study  ; 
therefore  Logic,  as  it  is  cultivated  at  the  present  day  must  be  a  fruitless 
study  likewise. 

58.  Repentance  is  a  good  quality  ;  wicked  men  abound  in  repentance, 
and  therefore  abound  in  what  is  good. 

59.  Warm  countries  aloue  produce  wine.  Spain  is  a  warm  country,  and 
therefore  produces  wine. 

60.  It  is  an  intensely  cold  climate  that  is  sufficient  to  freeze  mercury  ; 
the  climate  of  Siberia  is  sufficient  to  freeze  it,  and  hence  must  be  intensely 
cold. 

61.  No  designing  person  ought  to  be  trusted  ;  engravers  are  by  profes- 
sion designers  ;  therefore  they  ought  not  to  be  trusted. 

62.  I  will  not  do  this  act  because  it  is  unjust  ;  I  know  it  is  unjust  be- 
cause my  conscience  tells  me  so,  and  my  conscience  tells  me  so  because 
the  act  is  wrong. 

63.  Is  a  stone  a  body  ?  Yes.  Then  is  not  an  animal  a  body  ?  Yes. 
Are  you  an  animal  ?  I  think  so.  Ergo,  you  are  a  stone,  being  a  body. — 
Lucian. 


PRACTICAL   EXERCISES  387 

64.  If  ye  were  Abraham's  children  ye  would  do  the  works  of  Abraham. 
— John  viii.  39. 

65.  He  that  is  of  God  heareth  God's  words  ;  ye  therefore  hear  them  not, 
because  you  are  uot  of  God. — John  viii.  47. 

66.  His  imbecility  of  character  might  have  been  inferred  from  his 
proneness  to  favorites  ;  for  all  weak  princes  have  this  failing. 

67.  He  is  brave  who  conquers  his  passions  ,  he  who  resists  temptation 
conquers  his  passions  ;  so  that  he  who  resists  temptation  is  brave. 

68.  Suicide  is  not  always  to  be  condemned  ;  for  it  is  but  voluntary 
death,  and  this  has  been  gladly  embraced  by  many  of  the  greatest  heroes 
of  antiquity. 

69    All  that  glitters  is  not  gold  ;  tinsel  glitters  and  is  therefore  not  gold. 

70.  Meat  and  drink  are  the  necessaries  of  life.  The  revenues  of  the 
king  were  spent  on  meat  and  drink,  and  were  therefore  spent  on  the  nec- 
essaries of  life. 

71.  He  who  calls  you  a  man  speaks  truly  ;  he  who  calls  you  a  fool  calls 
you  a  man  ;  therefore  he  who  calls  you  a  fool  speaks  truly. 

72.  Theft  is  a  crime  ,  theft  was  encouraged  by  the  laws  of  Sparta ; 
therefore  the  laws  of  Sparta  encouraged  crime. 

73.  Since  all  gold  is  a  metal,  the  most  rare  of  all  masses  of  gold  must  be 
the  most  rare  of  all  the  metals. 

74.  Nothing  but  the  express  train  carries  the  mail,  and  as  the  last  train 
was  an  express  it  must  have  carried  the  mail. 

75.  Protective  laws  should  be  abolished,  for  they  are  injurious  if  they 
produce  scarcity,  and  they  are  useless  if  they  do  not. 

76.  The  Quaker  asserts  that  if  men  were  true  Christians  and  acted  upon 
their  religious  principles  there  would  be  no  need  of  armies ;  hence  he 
draws  the  conclusion  that  a  military  force  is  useless,  and  being  useless  is 
pernicious. 

77.  Detention  of  property  implies  at  least  possession  ;  for  detention  is 
natural  possession. 

78.  "Profit"  is  interpreted  or  defined  to  be  "advantage;"  to  take 
profit,  then,  is  to  take  advantage  ,  it  is  wrong  to  take  advantage  of  one's 
neighbor  ;  therefore  it  is  wrong  to  take  profit. 

79.  Peel's  remission  of  taxes  was  beneficial  ;  the  taxes  remitted  by  Peel 
were  indirect,  and  therefore  the  remission  of  indirect  taxes  is  beneficial. 

80.  Some  poisons  are  vegetable  ,  no  poisons  are  useful  drugs,  and 
therefore  some  useful  drugs  are  not  vegetable. 

81.  Whosoever  intentionally  kills  another  should  suffer  death  ;  a  soldier 
therefore  who  kills  his  enemy  should  suffer  death. 

82.  Few  towns  in  the  country  have  500,000  inhabitants,  and  since  all 
such  towns  ought  to  have  three  representatives  in  Congress,  it  is  evident 
that  few  towns  ought  to  have  three  representatives. 

83.  If  Bacon's  opinion  be  right  it  is  improper  to  stock  a  new  colony  with 


388  ELEMENTS  OF  LOGIC 

criminals  from  prison  ;  but  this  course  we  must  allow  to  be  proper  if  the 
method  of  colonizing  New  South  Wales  be  a  wise  one.  If  this  be  wise, 
therefore  Bacon's  opinion  is  not  right. 

84.  The  people  of  the  country  are  suffering  from  famine,  and  as  A,  B,  C 
are  people  of  the  country,  they  must  be  suffering  from  famine. 

85.  You  are  not  what  I  am  ;  I  am  a  man ;  therefore  you  are  not  a  man. 

86.  Gold  and  silver  are  wealth  ;  and  therefore  the  diminution  of  the 
gold  and  silver  of  a  country  by  exportation  is  the  diminution  of  the  wealth 
of  the  country. 

87.  The  holder  of  some  shares  in  a  lottery  is  sure  to  gain  a  prize,  and 
as  I  am  the  holder  of  some  shares  in  a  lottery  I  am  sure  to  gain  a  prize. 

88.  A  monopoly  of  the  sugar-refining  business  is  beneficial  to  sugar  re- 
finers ;  and  of  the  corn  trade  to  corn-growers  ;  and  of  the  silk  manufact- 
ure to  the  silk-weavers  ;  of  labor  to  the  laborers.  Now  all  these  classes 
of  men  make  up  the  whole  community.  Therefore  a  system  of  restric- 
tions upon  competition  is  beneficial  to  the  community. 

89.  Over-credulous  persons  should  never  be  believed;  and  as  the  ancient 
historians  were  in  many  instances  over-credulous  they  ought  never  to  be 
believed. 

90.  That  is  unfortunate  ;  you  insolently  assert  that  you  are  a  Darwinian, 
while  the  truth  is  that  you  are  a  poet. 

91.  Every  incident  in  the  narrative  is  probable,  and  hence  the  narrative 
may  be  believed  since  it  is  probable. 

92.  If  a  substance  is  solid  it  possesses  elasticity,  and  so  also  it  does  if  it 
be  liquid  or  gaseous  ;  but  all  substances  are  either  solid,  liquid,  or  gaseous  ; 
therefore  all  substances  possess  elasticity. 

93.  Who  is  most  hungry  eats  most  ;  who  eats  least  is  most  hungry  ; 
therefore  who  eats  least  eats  most. 

94.  If  the  Elixir  of  life  is  of  any  value  those  who  take  it  will  improve 
in  health  ;  now  my  friend  who  has  been  taking  it  has  improved  in  health, 
and  therefore  the  Elixir  is  of  value  as  a  curative  agent. 

95.  The  policy  of  protection  was  immediately  followed  by  a  great  in- 
crease in  the  prosperity  and  wealth  of  the  country,  and  hence  we  may 
infer  that  the  result  was  due  to  its  connection  with  the  enactment  of  the 
protective  law.  In  reply,  however,  we  are  told  that  before  the  passage  of 
trie  law  the  loss  by  fire  in  Chicago  in  one  year  was  $200,000,000,  but  was 
only  $3,000,000  for  the  year  after  its  passage,  so  great  was  the  effect  of 
this  act. 

96.  What  produces  intoxication  should  be  prohibited  ;  the  use  of  intox- 
icating liquors  causes  intoxication  ;  therefore  the  use  of  spirituous  liquors 
should  be  prohibited. 

97.  When  we  hear  that  all  the  righteous  people  are  happy,  it  is  hard  to 
avoid  exclaiming,  What !  are  all  the  unhappy  persons  we  see  thought  to 
be  unrighteous  ? 


PRACTICAL  EXERCISES  389 

98.  Italy  is  a  Catholic  country  and  abounds  in  beggars  ;  France  is  also 
a  Catholic  country,  and  therefore  abounds  in  beggars. 

99.  The  Latin  word  "virtus"  originally  meant  "manliness;"  hence 
the  virtue  of  manliness  or  courage  is  the  highest  virtue  and  type  of  all 
other  virtues. 

100.  If  it  be  fated  that  you  recover  from  your  present  disease,  you  will 
recover,  whether  you  call  in  a  doctor  or  not ;  again,  if  it  be  fated  that  you 
do  not  recover  from  your  present  disease,  you  will  not  recover,  whether 
you  call  in  a  doctor  or  not.  But  one  or  the  other  of  these  contradictories 
is  fated,  and  therefore  it  can  be  of  no  service  to  call  in  a  doctor. 

101.  This  person  may  reasonably  be  supposed  to  have  committed  the 
theft,  for  he  can  give  no  satisfactory  account  of  himself  on  the  night  of 
the  alleged  offence  ;  moreover  he  is  a  person  of  bad  character,  and,  being 
poor,  is  naturally  liable  to  a  temptation  to  steal. 

102.  All  the  trees  in  the  park  make  a  thick  shade;  this  oak-tree  is  one 
of  them  and  therefore  makes  a  thick  shade. 

103.  All  visible  bodies  shine  by  their  own  or  by  reflected  light.  The 
moon  does  not  shine  by  its  own  ;  therefore  it  shines  by  reflected  light ; 
but  the  sun  shines  by  its  own  light  ;  therefore  it  cannot  shine  by  reflected 
light. 

104.  The  two  propositions,  "Aristotle  is  living,"  and  "Aristotle  is 
dead,"  are  both  intelligible  propositions  ;  they  are  both  of  them  true  or 
both  of  them  false,  because  all  intelligible  propositions  must  be  either 
true  or  false. 

105.  How  can  anyone  maintain  that  paiu  is  always  an  evil  who  admits 
that  remorse  involves  pain,  and  yet  may  sometimes  be  a  real  good  ? 

106.  I  am  charged  with  absenteeism  from  my  post,  and  on  that  ground 
I  am  accused  of  ignorance  in  regard  to  the  proper  duties  of  my  office. 
But  my  accuser  himself,  who  was  my  predecessor  in  the  same  office,  was 
not  in  the  country,  of  which  he  was  the  ruler,  longer  than  five  days. 

107.  Every  law  is  either  useless  or  it  occasions  hurt  to  some  person  ; 
now  a  law  that  is  useless  ought  to  be  abolished  ;  and  so  ought  every  law 
that  occasions  hurt ;  therefore  every  law  ought  to  be  abolished. 

108.  What  fallacies  are  implied  or  charged  against  Sir.  Spencer  in  the 
following  criticism? 

"Mr.  Spencer's  distinction  between  objects  and  relations  is  far  from 
satisfactory;  and  even  if  it  were  a  true  distinction,  I  do  not  see  that  any 
adequate  classification  of  knowledge  could  be  based  upon  it,  because  there 
is  no  science  within  the  circle  of  knowledge  that  does  not  deal  both  with 
objects  and  relations." — Knight:  Essays  in  Philosophy. 

109.  Does  a  grain  of  millet,  when  dropped  on  the  floor,  make  sound  ? 
No.  Does  a  bushel  of  millet  make  sound  under  the  same  circumstances  ? 
Yes.  Is  there  not  a  determinate  proportion  between  the  bushel  and  the 
grain  ?     There  is.     There  must  therefore  be  the  same  proportion  between 


390  ELEMENTS  OF  LOOIO 

the  sonorousness  of  the  two.     If  one  grain  be  not  sonorous,  neither  can 
ten  thousand  grains  be  bo. 

1 10.  What  you  say  is  that  virtue  is  the  power  of  attaining  good  ?  Yes. 
And  you  would  say  that  goods  are  such  as  health  and  wealth,  and  the 
possession  of  gold  and  silver,  and  having  office  and  honor  in  the  state — 
these  are  whal  von  call  goods  ?  Yes,  all  these.  Then,  according  to 
IWcno,  who  is  the  hereditary  friend  of  the  great  king,  virtue  is  the  power 
of  getting  Bilver  and  gold.  —  Plato's  Dialogues  :  Meno. 

111.  Injustice  is  more  profitable  than  justice  because  those  who  do  un- 
just acts  gain  more  than  the  just. 

112.  As  for  saying  that  without  God  man  cannot  have  moral  sentiments, 
or.  in  other  words,  cannot  distinguish  between  vice  and  virtue,  it  is  as  if 
one  said  that  without  the  idea  of  God  man  would  not  feel  the  necessity 
of  eating  and  drinking. — John  Morley. 

113.  I  am  offered  a  sum  of  money  to  assist  this  person  in  gaining  the 
office  he  desires;  to  assist  a  person  is  to  do  him  good,  and  no  rule  of 
morality  forbids  tie-  doing  of  good  ;  therefore  no  rule  of  morality  forbids 
me  to  receive  the  sum  of  money  for  assisting  this  person. 

114.  Ruminant  animals  are  those  which  have  cloven  feet,  and  they 
usually  have  horns  ;  the  extinct  animal  which  left  this  foot-print  had  a 
cloven  foot ;  therefore  it  was  a  ruminant  animal  and  had  horns.  Again, 
as  no  beasts  of  prey  are  ruminant  animals,  it  cannot  have  been  a  beast  of 
prey. 

115.  Without  order  there  is  no  living  in  public  society,  because  the 
want  thereof  is  the  mother  of  confusion,  whereupon  division  of  neces- 
sity followeth.  and  out  of  division  destruction. — Hooker:  Ecclesiastical 
Polity. 

116.  The  man  who  does  any  kind  of  work  in  a  careless,  bungling,  or 
superficial  way  is  not  acting  as  a  reasonable  being  ;  for  the  first  demand 
of  reason,  as  the  truthful  faculty  in  the  world  of  action,  is  to  realize  its 
idea  completely  and  thoroughly,  and  this  no  hasty  and  superficial  handi- 
work will  pretend  to  do. 

117.  Happiness  signifies  a  gratified  state  of  all  the  faculties.  The  grati- 
fication of  a  faculty  is  produced  by  its  exercise.  To  be  agreable  that  exer- 
cise must  be  proportionate  to  the  power  of  the  faculty  ;  if  it  is  insufficient 
discontent  arises  and  its  excess  produces  weariness.  Hence  to  have  com- 
plete felicity  is  to  have  all  the  faculties  exerted  in  the  ratio  of  their  sev- 
eral developments. 

118.  We  must  either  gratify  our  vicious  propensities  or  resist  them  ; 
the  former  course  will  involve  us  in  sin  and  misery  ;  the  latter  requires 
self-denial.  Therefore  we  must  either  fall  into  sin  and  misery,  or  practise 
self-denial. 

119.  Every  moral  aim  requires  the  rational  means  of  attaining  it  ;  these 
means  are  the  establishment  of  laws ;  and  as  happiness  is  the  moral  aim  of 


PRACTICAL  EXERCISES  301 

man  it  follows  that  the  attainment  of  it  requires  the  establishment  of 
laws. 

120.  He  that  can  swim  needs  not  despair  to  fly  ;  for  to  swim  is  to  fly  in 
a  grosser  fluid,  and  to  fly  is  to  swim  in  a  subtler  fluid. 

121.  What  fallacy  was  the  humorist  afraid  of  who  said  that  he  would 
not  admit  that  two  and  two  make  four  until  he  knew  what  use  was  to  be 
made  of  the  assertion. 

122.  The  Good  is  a  state  of  consciousness  ;  for  the  Good  is  a  possible  ob- 
ject of  knowledge  ;  but  all  objects  of  knowledge  are  states  of  consciousness. 
Hence  the  Good  is  a  state  of  consciousness. 

123.  The  Good  is  pleasure  ;  for  the  Good  results  from  the  due  perform- 
ance of  function  ;  but  the  Good  is  a  state  of  consciousness  ;  therefore  the 
Good  is  the  state  of  consciousness  which  results  from  the  due  performance 
of  function. 

124.  Riches  are  for  spending,  and  spending  for  honor  and  good  actions; 
therefore  extraordinary  expense  must  be  limited  by  the  worth  of  the  oc- 
casion. 

125.  The  several  species  of  brutes  being  created  to  prey  upon  one  an- 
other proves  that  the  human  species  were  intended  to  prey  upon  them. 

126.  If  any  objection  can  be  urged  to  justify  a  change  of  established 
laws,  no  laws  could  be  reasonably  maintained  ;  but  some  laws  can  be 
reasonably  maintained  ;  therefore  no  objection  that  can  be  urged  will 
justify  a  change  of  established  laws. 

127.  You  are  inconsistent  with  yourself,  for  you  told  me  yesterday  that 
there  was  a  presumption  of  this  man's  guilt,  and  now  when  I  say  that  I 
may  presume  his  guilt,  you  contradict  me. 

128.  The  more  correct  the  logic,  the  more  certainly  the  conclusion  will 
be  wrong  if  the  premises  are  false  ;  therefore  where  the  premises  are 
wholly  uncertain  the  best  logician  is  the  least  safe  guide. 

129.  If  our  rulers  could  be  trusted  always  to  look  to  the  best  interests 
of  their  subjects  monarchy  would  be  the  best  form  of  government ;  but 
they  cannot  be  trusted  ;  therefore  monarchy  is  not  the  best  form  of 
government. 

130.  He  who  bears  arms  at  the  command  of  the  magistrate  does  what 
is  lawful  for  a  Christian  ;  the  Swiss  in  the  French  service,  and  the  Britisli 
in  the  American  service,  bore  arms  at  the  command  of  the  magistrate  ; 
therefore  they  did  what  was  lawful  for  a  Christian. 

131.  A  man  that  hath  no  virtue  in  himself  envieth  virtue  in  others ;  for 
men's  minds  will  either  feed  upon  their  own  good  or  upon  others  evil,  and 
who  wanteth  the  one  will  prey  upon  the  other. 

132.  The  object  of  war  is  durable  peace  ;  therefore  soldiers  are  the  best 
peacemakers. 

133.  Confidence  in  promises  is  essential  to  human  intercourse  and  com- 
merce ;  for  without  it  the  greatest  part  of  our  conduct  would  proceed  upon 


392  ELEMENTS   OF  LOCK' 

chanoe.  But  there  could  be  no  confidence  in  promises  if  men  were  noi 
,,i,i,  ed  to  perform  them;  the  obligation,  therefore,  to  perform  promises 
is  essential  to  the  same  ends  .-111(1  in  the  Bame  d< 

184,  [f  the  majority  of  those  who  use  publio-houseE  are  prepared  to 
close  them  legislation  is  unnecessary;  but  if  they  are  not  prepared  for 
such   a  measure,  then  to  force  it  on  them  by  outside  pressure  is  both 

.Ian    .runs  ami  unjust. 

135.  Se  who  believes  himself  to  be  always  in  the  right  in  his  opinion 
lavs  claim  to  infallibility  ;  you  always  believe  yourself  to  be  in  the  right 
in  pour  opinion ;  therefore  yon  lay  claim  to  infallibility. 

186.  If  we  never  find  skins  except  as  integuments  of  animals  we  may 
safely  conclude  that  animals  cannot  exist  without  skins.  If  color  can- 
not exist  by  itself,  it  follows  that  neither  can  anything  that  is  colored  ex- 
ist without  color.  So,  if  language  without  thought  is  unreal,  thought 
without  language  must  also  be  so. 

137.  If  the  light  is  not  refracted  near  the  surface  of  the  moon  there 
cannot  be  any  twilight;  but  if  the  moon  has  do  atmosphere  light  is  not 
refracted  near  its  Burfaoe;  therefore  if  the  moon  has  no  atmosphere 
there  cannot  be  any  twilight 

138.  No  soldiers  should  be  brought  into  the  field  who  are  not  well  qual- 
ified to  perform  their  duty  ;  none  but  veterans  are  well  qualified  to  per- 
form their  part ;  therefore  none  but  veterans  should  be  brought  into  the 
field. 

139.  The  minimum  >  '"  is  the  least  magnitude  which  can  be  seen  ; 
no  part  of  it  alone  is  visible,  and  yet  all  parts  of  it  must  affect  the  mind 
in  order  that  it  may  be  visible  ;  therefore  every  part  of  it  must  affect  the 
mind  without  being  visible. 

140.  Improbable  events  happen  almost  every  day,  but  what  happens 
almost  every  day  is  a  very  probable  event ;  therefore  improbable  events 
are  very  probable  events. 

141.  What  fallacies  are  implied  against  an  opponent  in  the  following 
statement :  "  Each  of  its  links  is  in  fact  unSound.  And  even  though  no 
flaw  were  visible  in  them,  still  the  conclusion  is  demonstrably  false." 

142.  "  Xow  that  which  does  not  make  a  man  worse,  how  can  it  make 
a  man's  life  worse  ?  But  neither  through  ignorance,  nor  having  the 
knowledge  but  not  the  power  to  guard  against  or  correct  these  things,  is 
it  possible  that  the  nature  of  the  universe  has  overlooked  them :  nor  is  it 
possible  that  it  has  made  so  great  a  mistake,  either  through  want  of  power 
or  want  of  skill,  that  good  and  evil  should  happen  indiscriminately  to  the 
good  and  the  bad.  But  death  certainly,  and  life,  honor  and  dishonor, 
pain  and  pleasure — all  these  things  happen  equally  to  good  men  and  bad, 
being  things  which  make  us  neither  better  nor  worse.  Therefore  they 
are  neither  good  nor  evil." — Marcus  Aurdius. ' 

143.  Since  there  is  no  harm  or  evil  to  the  elements  themselves  in  their 


PRACTICAL  EXERCISES  393 

continual  changes  into  one  another,  a  man  should  have  no  apprehension 
about  the  dissolution  of  all  elements.  For  it  is  according  to  nature,  and 
nothing  is  evil  that  is  according  to  nature. — Marcus  Aurelius. 

144.  Form  a  syllogism  and  show  under  what  conditions  any  one  of  the 
following  three  fallacies  may  be  found — non  sequitur,  petitio  princijrii, 
and  equivocation. 

145.  What  fallacy  is  charged  to  the  defenders  of  Charles  the  Second  by 
Macaulay  in  the  following  statements:  "We  charge  him  with  having 
broken  his  coronation  oath,  and  we  are  told  that  he  kept  his  marriage 
vow  !  We  accuse  him  of  having  given  up  his  people  to  the  merciless  in- 
flictions of  the  most  hot-headed  and  hard-hearted  of  prelates,  and  the  de- 
fence is  that  he  took  his  little  son  on  his  knee  and  kissed  him  !  We  cen- 
sure him  for  having  violated  the  articles  of  the  Petition  of  Right,  after 
having  for  good  and  valuable  consideration  promised  to  observe  them, 
and  we  are  informed  that  he  was  accustomed  to  hear  prayers  at  six  o'clock 
in  the  morning." 

140.  "Don't  you  think  the  possession  of  gold  is  good?  Yes,  said 
Ctesippus,  and  the  more  the  better.  And  to  have  money  everywhere  and 
always  is  a  good  ?  Certainly,  a  great  good,  he  said.  And  you  admit  that 
gold  is  a  good  ?  I  have  admitted  that,  he  replied.  And  ought  not  a  man 
then  to  have  gold  everywhere  and  always,  and  as  much  as  possible  in 
himself,  and  may  not  he  be  deemed  the  happiest  of  men  who  has  three 
talents  of  gold  in  his  stomach,  and  a  talent  in  his  head,  and  a  stater  of 
gold  in  either  eye." — Plato's  Dialogues :  Euthydemus. 

147.  "  If  we  are  to  test  the  truth  of  materialism  by  its  outcome  for 
well-being,  we  can  hold  it  only  by  showing  that  the  supreme  end  of  man 
is  to  develop  a  body,  and  that  materialism  is  especially  useful  in  promot- 
ing the  interests  of  the  animal  nature.  The  normal  brain  is  that  which 
takes  care  of  itself,  and  the  test  of  truth  is  self-preservation.  Moral  aims 
and  scientific  truth,  so  far  as  they  have  no  physical  value,  must  be  voted 
not  merely  worthless,  but  delusion  ;  for  the  test  of  truth  is  physical  pres- 
ervation. Hence  the  inhabitant  of  the  sty  would  be  the  prince  of  ma- 
terialistic philosophers ;  he  is  not  troubled  by  delusion  and  he  preserves 
himself." 

143.  If  sin  by  itself  confers  the  right  and  imposes  the  duty  of  punish- 
ment, there  must  be  the  right  to  inflict  either  a  definite  punishment  or 
an  infinite  amount.  If  the  latter,  it  is  obvious  that  the  state  will  always 
have  the  right  to  inflict  any  quantity  of  punishment  it  pleases  upon  any 
of  its  citizens  at  any  time,  since  all  have  sinned  and  incurred  thereby  an 
unlimited  liability  to  punishment.  If,  on  the  other  hand,  wrong-doing 
confers  a  right  to  inflict  a  merely  limited  amount  of  punishment,  it  will 
not  be  possible  to  determine  the  amount  outside  of  utilitarian  considera- 
tions, since  moral  guilt  cannot  be  measured  in  terms  of  physical  pain. 
But  it  is  apparent  that  the  right  to  inflict    an  infinite  punishment  with- 


394  ELEMENTS  OF  LOGIC 

out  distinction  of  the  crimes  in  regard  to  consequences  is  absurd,  as  also 
the  infliction  of  a  definite  amount  without  regard  to  its  utility,  and  hence 
sin  by  itself  and  independently  of  the  advantage  to  society  is  not  punisha- 
ble. 

149.  Our  tariff  is  found  fault  with  because  it  does  not  make  men  inde 
pendent  and  virtuous,  besides  giving  them  the  opportunity  to  become 
prosperous.  It  is  said  to  be  responsible  for  the  over  production  which 
has  characterized  some  branches  of  manufacture.  The  same  evil  occurs, 
and  more  frequently,  in  Lancashire  under  Free  Trade. 

150.  The  usefulness  of  government  has  been  established  by  a  long  ex- 
perience in  the  enactment  and  enforcing  of  laws  against  such  acts  as  in- 
jure social  order,  and  if  anything  be  needed  to  establish  the  benefits  of 
despotic  governments  it  can  be  found  in  the  power  and  practice  exercised 
by  them  to  punish  crimes  against  life  and  property.  Therefore  we  may 
infer  their  usefulness  as  forms  of  government. 

151.  If  a  debater  affirm  a  proposition  in  the  major  premise  which  is 
true  only  in  an  abstract  sense  ;  that  is,  of  the  genus  or  conferentia,  and 
the  minor  premise  expresses  what  is  true  of  unessential  properties,  what 
fallacies  would  be  implied,  first,  by  indicating  this  difference  ;  second,  by 
disputing  the  universality  of  the  major  premise,  if  the  argument  was  in 
the  first  Figure  ;  and  third,  by  disputing  the  truth  of  either  premise. 

152.  "  Five  years  ago  a  first-class  pair  of  nickel-plated  steel  skates, 
with  the  necessary  clamps  to  fasten  them  to  the  boot  or  shoe,  cost  $15. 
To-day  precisely  the  same  article,  and  with  an  equal  finish  and  complete- 
ness, can  be  obtained  for  $4.  Three  years  ago  a  second  grade  of  nickel- 
plated  steel  skates  cost  $4.  The  same  article  can  be  produced  to-day  for 
$1.50.  The  decline  of  seventy  per  cent,  in  five  years,  and  of  sixty  percent, 
in  three  years,  shows  just  how  protection  cheapens  prices." — Milwaukee 
Evening  Wisconsin. 

153.  If  the  earth  were  of  equal  density  throughout,  it  would  be  abouf 
21  times  as  dense  as  water  ;  but  it  is  about  5$  times  as  dense ;  therefore 
the  earth  must  be  of  unequal  density. 

154.  "  '  By  open  discrimination,  or  by  secret  rates,  drawbacks,  and  re- 
bates, a  few  railway  managers  may  subject  to  their  will  every  business  in 
which  transportation  is  a  large  element  of  cost,  as  absolutely  as  any  Ori- 
ental despot  ever  controlled  the  property  of  his  subjects.  No  civilized 
community  has  ever  known  a  body  of  rulers  with  such  power  to  distribute 
at  pleasure,  among  its  mercantile  classes,  prosperty  or  adversity,  wealth 
or  ruin.  That  this  is  no  abstract  or  remote  danger  to  society  is  plain  to 
any  man  who  will  look  at  the  condition  of  trade  and  of  mercantile  morals 
in  the  United  States  to-day.'  How  vivid!  But  how  absurd !  how  un- 
true !  Our  commercial  morals  are  equal  to  the  highest  in  the  world." — 
Kirkman:  Railway  Bates  and  Government  Control. 

155.  "Not  only  the  effects  are  good,  but  the  agent  sees  beforehand 


PRACTICAL  EXERCISES  395 

that  they  will  be  so.  This  may  make  the  action  indeed  (done  from  an- 
tipathy) a  perfectly  right  action  ;  but  it  does  not  make  antipathy  a  right 
ground  of  action.  For  the  same  sentiment  of  antipathy,  if  implicitly  de- 
ferred to,  may  be,  and  very  frequently  is,  productive  of  the  very  worst 
effects.  Antipathy,  therefore,  can  never  be  a  right  ground  of  action." — 
Bentham  :  Principled  of  Morals  and  Legislation. 

156.  "Mr.  Gladstone,  however,  commits  himself  to  the  principle  that 
'  all  protection  is  morally  bad.'  If  this  has  been  his  belief  ever  since  he 
became  an  advocate  of  free  trade,  his  conscience  must  have  received 
many  and  severe  wounds,  as  session  after  session,  while  Chancellor  of  the 
Exchequer,  he  carried  through  Parliament  a  bounty — may  I  not  say  a  di- 
rect protection  ? — of  £180,000  to  a  line  of  steamers  running  between  Eng- 
land and  the  United  States — a  protection  that  began  six  years  before  free- 
trade  was  proclaimed,  and  was  continued  nearly  twenty  years  after." — Mr. 
Blaine,  in  tlie  North  American  Review  for  January,  1890. 


DEDUCTIVE  AND  INDUCTIVE 

Examine  the  following  arguments,  stating  whether  they  are  deductive 
or  inductive  ;  if  deductive,  show  whether  they  are  valid  or  invalid,  and 
why  ;  if  inductive,  show  what  Method  of  Induction  is  involved. 

1.  Two  of  the  wealthiest  men  of  the  West  are  said  to  have  been  mes- 
senger-boys.    It  pays  to  go  slow,  after  all. 

2.  Geometry  contemplates  figures.  Figure  is  the  termination  of  mag- 
nitude ;  but  extension  in  the  abstract  has  no  definite  determinate  magni- 
tude. Whence  it  follows  clearly  that  it  can  have  no  figure,  and  conse- 
quently is  not  the  object  of  Geometry,  whose  object  is  commonly  said  to 
be  abstract  extension. 

3.  The  newly  discovered  painting  must  be  a  Rubens  ;  for  the  concep- 
tion, the  drawing,  the  tone  and  the  tints  are  precisely  those  seen  in  the 
authentic  works  of  that  master. 

4.  In  nine  counties,  in  which  the  population  is  from  100  to  150  per 
square  mile,  the  births  to  100  marriages  are  396  ;  in  sixteen  counties, 
with  a  population  of  150  to  200  per  square  mile,  the  births  are  390  to  100 
marriages.  Therefore  the  number  of  births  per  marriage  is  inversely  re- 
lated to  the  density  of  population,  and  contradicts  Malthus's  theory  of  the 
law  of  population. 

5.  "Cramming  "  for  examination  is  detrimental  rather  than  otherwise  ; 
for  I  have  noticed  that  no  matter  what  the  subject  is,  I  invariably  write 
a  poor  paper  when  I  "  cram,"  and  a  good.one  when  I  do  not. 

6.  The  great  famine  in  Ireland  began  in  1845,  and  increased  until  it 
reached  a  climax  in   1848.     During  this  time  agrarian  crime  increased 


396  ELEMENTS  OF  LOGIC 

very  rapidly  until  in  1848  it  was  more  than  three  times  as  great  as  in 
1845.  After  this  time  it  decreased  with  the  return  of  better  crops,  until 
in  1851  it  was  only  fifty  per  cent,  more  than  it  was  in  1845.  It  is  evident 
from  this  that  a  close  relation  of  cause  and  effect  exists  between  famine 
and  agrarian  crime. 

7.  Pitt  did  not  bribe  the  Irish  parliament  in  1800,  when  he  so  lavisbly 
bestowed  peerages  on  its  members.  For  he  bestowed  honors  on  only 
forty  of  his  followers,  while  in  1779,  Lord  North  bestowed  thirty  peerages 
in  one  day,  and  in  1832,  Lord  Grey  got  the  king's  consent  to  tbe  creation 
of  a  hundred. 

8.  "Suppose  we  have  a  southward  velocity  amounting,  let  us  say,  to 

3  feet  per  second,  and  simultaneously  an  eastward  velocity  amounting  to 

4  feet  per  second,  then  we  know  by  kinematics  how  to  construct  the 
single  velocity  which  is  the  resultant  of  these  two.  All  we  have  to  do  is 
to  draw  a  line  of  length  3  southward  and  from  its  extremity  a  line  of 
length  4  to  the  eastward,  and  then  complete  the  triangle.  In  a  geometri- 
cal sense,  therefore,  a  velocity  of  3  southward  and  a  velocity  of  4  east- 
ward will  be  equivalent  to  a  velocity  which,  if  you  calculate  what  the 
third  side  of  that  triangle  will  be,  is  represented  by  5  on  that  scale." — 
Recent  Advances  in  Physical  Science. 

9.  On  May  27,  1875,  a  remarkable  shower  of  small  pieces  of  hay  oc- 
curred at  Monkstown,  near  Dublin.  They  appeared  floating  slowly  down 
from  a  great  height.  A  similar  shower  occurred  a  few  days  earlier  in 
Denbighshire,  from  this  and  many  similar  facts  we  may  conclude  that 
the  distribution  of  organisms  of  the  same  species  over  continents  and 
islands  separated  by  the  ocean  has  been  effected  by  the  agency  of  natural 
forces. 

10.  The  influence  of  heat  in  changing  the  level  of  the  ground  upon 
which  the  Teniple  of  Jupiter  Serapis  stands  might  be  inferred  from 
several  circumstances.  In  the  first  place,  there  are  numerous  hot  springs 
in  the  vicinity,  and  when  we  reflect  on  the  dates  of  the  principal  oscilla- 
tions of  level  this  conclusion  is  made  much  more  probable.  Thus  before 
the  Christian  era,  when  Vesuvius  was  regarded  as  a  spent  volcano,  the 
ground  upon  which  the  temple  stood  was  several  feet  above  water.  But 
after  the  eruption  of  Vesuvius  in  79  B.C.  the  temple  was  sinking.  Subse- 
quently Vesuvius  became  dormant  and  the  foundations  of  the  temple 
began  rising.  Again  Vesuvius  became  active,  and  has  remained  so  ever 
since.  During  this  time  the  temple  has  been  subsiding  again,  so  far  as 
we  know  its  history.  , 

11.  "I  have  two  pendulums  with  very  massive  bobs  suspended  from 
them,  and  have  carefully  made  these  two  pendulums  as  nearly  as  possi- 
ble the  same.  I.otli  pendulums  are  now  at  rest,  but  suppose  1  set  one  to 
vibrate,  leaving  the  other  at  rest,  you  will  notice,  if  you  watch  the  second 
for  a  short  time  that  it  begins  to  vibrate  in  its  turn,  and  as  time  goes  on 


PRACTICAL  EXERCISES  397 

it  swings  through  larger  and  larger  arcs  of  vibration  till  at  last  the  first 
pendulum  is  brought  to  rest.  Jsow  this  is  quite  obviously  a  case  of  trans- 
ference of  energy  from  one  pendulum  to  the  other,  effected,  you  will  see, 
through  the  wooden  structure." — Recent  Advances  in  Physical  Science. 

12.  Why  should  any  but  professional  moralists  trouble  themselves  with 
the  solution  of  moral  difficulties  ?  For,  as  we  resort  to  a  physician  in 
case  of  any  physical  disease,  so,  in  case  of  any  moral  doubt  or  any  moral 
disorganization,  it  seems  natural  that  we  should  rely  on  the  judgment  of 
some  man  specially  skilled  in  the  treatment  of  such  subjects. 

13.  Take  a  bottle  of  soda-water,  slightly  warmer  than  a  given  tempera- 
ture registered  by  the  thermopile,  and  mark  the  deflection  it  causes.  Then 
cut  the  string  which  holds  it,  and  the  cork  will  be  driven  out  by  the 
elastic  force  of  the  carbonic  acid  gas.  The  gas  performs  its  work,  and  in 
so  doing  it  consumes  heat,  and  the  deflection  of  the  thermopile  shows 
that  the  bottle  is  cooler  than  before,  heat  having  been  lost  in  the  pro- 
cess. 

14.  The  occurrence  of  the  Aurora  Borealis  under  different  meteorolog- 
ical conditions  is  invariably  accompanied  by  magnetic  disturbances  and 
by  the  appearance  of  sun  spots,  and  hence  we  infer  that  a  causal  connec- 
tion exists  between  them  and  the  sun  spots. 

15.  It  has  been  found  that  linnets  when  shut  up  and  educated  with 
singing  larks — the  skylark,  woodlark,  or  titlark — will  adhere  entirely  to 
the  songs  of  these  larks  instead  of  the  natural  song  of  the  linnets.  We 
may  infer,  therefore,  that  birds  learn  to  sing  by  imitation,  and  that  their 
songs  are  no  more  innate  than  language  is  in  man. 

16.  An  enemy  has  a  keener  perception  than  a  friend  ;  for,  as  Plato 
says,  the  "lover  is  blind  as  respects  the  loved  one,"  and  hatred  is  both 
curious  and  gossipy.  Hiero  was  twitted  by  one  of  his  enemies  for  the 
foulness  of  his  breath  ;  so  he  went  home  and  said  to  his  wife  :  "  How  is 
this  ?  You  never  told  me  of  it."  But  she,  being  pure  and  innocent,  re- 
plied :  "I  thought  all  mens  breath  was  like  that."  Thus  perceptible 
and  material  things,  and  things  that  are  plain  to  everybody,  are  sooner 
learned  from  enemies  than  from  friends.—  Plutarch's  Men/Is. 

17.  A  man  cannot  really  be  injured  by  his  brethren,  for  no  act  of  theirs 
can  make  him  bad,  and  he  must  not  be  angry  with  them  nor  hate  them  ; 
for  we  are  made  for  co-operation,  like  feet,  like  hands,  like  eyelids,  like 
the  rows  of  the  upper  and  lower  teeth.  To  act  against  one  another,  then, 
is  contrary  to  nature  ;  and  it  is  acting  against  one  another  to  be  vexed 
and  to  turn  away.  —  Thoughts  of  Afaivua  Aurelius. 

18.  As  an  evidence  of  the  remote  antiquity  of  highly  civilized  man  we 
have  the  following  facts:  On  one  of  the  remote  islands  of  the  Pacific-  - 
Easter  Island— two  thousand  miles  from  South  America,  two  thousand 
miles  from  the  Marquesas,  and  more  than  one  thousand  miles  from  the 
Gambier  Islands,    are   found  hundreds    of   gigantic  stone  images,    now 


39S  ELEMENTS  OF  LOGIO 

mostly  in  ruins.  They  are  often  forty  feet  high,  while  some  seem  to 
have  been  much  larger,  the  crowns  on  their  beads,  out  out  of  a  red  stone, 
being  sometimes  ten  feet  in  diameter,  while  even  the  head  and  neck  of 
one  is  said  to  bave  been  twenty  feet  high.  The  island  containing  these 
remarkable  works  has  an  area  of  about  thirty  square  miles,  and  as  the 
smallest  image  is  about  eight  feet  high,  weighing  four  tons,  and  as  the 
Largest  must  weigh  over  a  hundred  tons  or  much  more,  th.-ir  existence 

implies  a  large  population,  ahundance  of  food,  aud  an  established  gov- 
ernment which  so  small  an  island  could  not  Bupply. 

19  We  observe  very  frequently  that  very  poor  handwriting  character- 
izes the  manuscripts  of  able  men,  while  the  best  handwriting  is  as  fre- 
quent with  those  who  do  little  mental  work  when  compared  with  those 
whose  penmanship  is  poor.  We  may,  therefore,  infer  that  poor  penman- 
ship is  caused  by  the  iniluence  of  severe  mental  occupation. 

20.  It  has  been  shown  by  observation  that  overdriven  cattle,  if  killed 
before  recovery  from  their  fatigue,  become  rigid  and  putrefy  in  a  sur 
prisingly  short  time.  A  similar  fact  has  been  observed  in  the  case  of  ani- 
mals hunted  to  death,  cocks  killed  during  or  Bhortly  after  a  fight,  and 
soldiers  slain  in  battle.  The  contrary  is  remarked  when  the  muscular  ex- 
ercise has  not  been  great  or  excessive.  Eence  we  may  infer  that  cada- 
veric rigidity  depends  upon  a  more  or  less  unirritable  condition  of  the 
muscles  immediately  before  death. 


INDEX 


Absolute  terms,  47-49 

Abstract  terms,  36—44,  65  ;  relation  to 
fallacy  of  accident,  235 

Abstraction,  26,  40 

Accent,  fallacy  of,  220,  221 

Accidens,  82 

Accident,  82-86  ;  fallacies  of,  231-240 

Accidentia,  88 

Acquisition,  341 

Affirmative  propositions,  109 

Agreement,  principle  of,  331,  332 ; 
method  of,  354 

All,  logical  signification  of,  111 

Alternative,  109,  212 

Abigaity  of  terms,  50-67  ;  of  proposi- 
tions, 115-121 

Ambiguous  middle,  226,  239 

Ampliative  propositions,  112 

Analytic  propositions,  112 

Antecedent,  108,  204 

Antithesis,  169 

Any,  logical  signification  of,  111 

Argumentum  ad  rem,  250 ;  ad  judi- 
cium, 250  ;  ad  populum,  251  ;  ad 
hominem,  251 ;  ad  ignorantiam,  251  ; 
ad  verecundiam,  251 

Aristotle,  248,  264,  267 

Attributive  terms,  37 

Bacon,  298 
Bain,  32 

Barbara,  Celarent,  etc.,  190 
Begging  the  question,  240 
Benjamin  Franklin,  306 
Bentham,  Jeremy,  243 
Bentham,  George,  262 
Berkeley,  250 
Brewster,  Sir  David,  332 
26 


Categorematic  terms,  31 

Categorical  propositions,  107 ;  reason- 
ing, 204 

Categories,  82 

Cause,  fallacy  of  false,  255 

Circulus  in  definiendo,  104 

Circulus  in  probando,  241,  244 

Classification,  343 ;  of  fallacies,  219-227 

Cognition,  22 

Collective  terms,  34,  35 

Comparison,  23 

Complimentary  propositions,  116 

Complex  syllogisms,  197 

Composition,  fallacy  of,  228 

Comprehension.     See  Intension 

Concepts  and  conceptions  defined,  20  ; 
formation  of,  21-27 ;  kinds  of,  25- 
26,  31-49 ;  denomination  of,  26 ;  in- 
tension and  extension  of,  68-81 ; 
analysis  of,  94  ;  infinitated,  165 

Conception,  5,  16,  19-27,  28 

Conclusion,  29,  171 

Conclusion,  weakened,  184 

Concomitant  variations,  method  of,  357 

Concrete  terms,  36-44  ;  relation  to  fal- 
lacy of  accident,  235 

Conditional  propositions,  107 

Conferentia,  87,  88,  91,  102,  126,  127, 
234,  235,  278,  323 

Confusion  of  fallacies,  256-261 

Connotation  of  terms,  79 

Connotative,  meaning  of,  79-81 

Consequent,  108,  204  ;  fallacy  of  false, 
253 

Contradiction,  law  of,  291 

Contradictory,  142 

Contraposition.  See  Contraversion, 
note  on,  163 


400 


INDEX 


Contraries,  142,  144 

Controversion,  ltil-160 

Contribution,  His 

Converse,  L56 

Converse  fallacy  of  accident,  231 

Conversion,  L55-160;  Bimple,  156,264; 
per  aocidens  or  limited,  L56  ;  roles 
for,  150;  by  negation,  159  j  table  of, 
L60 

Convertend,  L56 

Co-ordinate  species.  96 

Deduction,  279,  .300,  301,  327,  358 
Dedactive  reasoning,  1">,  279,  295,  327, 
328 

Deductive  method,  337 

Definite  propositions,  110;  quantity, 
267 

Definition,  100-104;  kinds  of,  100; 
logical,  101  ;  rules  for,  103  ;  in  meth- 
od, 337 

Definition  and  division,  S2-104 

De  Morgan,  233,  338,  243,  -.'17.  252,  255, 
262 

Denomination  of  concepts,  26 

Denotation  and  connotation  of  terms, 
78 

Denotative,  meaning  of,  79-81 

Descriptive  definition,  100 

Destructive  dilemma,  217  ;  hypotheti- 
cal reasoning,  205 

Desynonymization  of  terms,  60 

Difference,  principle  of,  331,  333  ; 
method  of,  355 

Differentia,  82,  Sfi,  90 

Differential  or  specific  accident,  fallacy 
of,  231 

Differentiation  of  terms,  60 

Dilemma,  217  ;  constructive,  217  ;  de- 
structive, 217 

Dilemmatic  propositions,  108 

Direct  reduction,  190 

Disjunction,  incomplete,  214 

Disjunctive  propositions,  107 ;  syllo- 
gisms, 212-218 

Distribution  of  terms,  138 ;  rules  for, 
140 

Distributive  terms,  34,  35 


Division,  definition  and,  82-104;  logi 
cal,  91  ;  rules  for,  95;  dichotomoue 
97 ;  triohotomona,  97 

Diviaionia,  fnndamentnm,  95 

Dr.  Johnson 

Daplea  proposil  ions,  1 L6 

Doctrini  of  logical,  16-30 

EWTHYMEME,  198 

Epicheirema,  200 
Episyllogiam,  L98 

Jvpii vocal  terms,  51 

Eqni vocation,  fallacies  of,    225, 
240 

Essence,  86 

Essentia,  85,  s^ 

Essential  properties,  84 ;  propositions, 
L12 

Ethics,  14 

Etymological  definition,  100 

Euler,  130 

Exclamatory  propositions,  107 

Exceptive  propositions,  120 

Excluded  middle,  law  of,  1  15 

Exclusive  propositions,  118  ;  quantity 
of,  119 

Experiment,  342,  352 

Explicative  propositions,  112 

Extension  of  concepts,  68-81  ;  relation 
between  intension  and,  72  ;  law  of, 
and  intension,  73 

Extension,  judgments  of,  123 ;  sym- 
bolized, 130,  131 


Fallacia  consequentis,  253 

Fallacies,  classification  of,  219-227  : 
hermeneutic,  220;  formal  or  logical, 
223 ;  material,  225.  228-261  ;  of  am- 
phibology, 220 ;  of  accent,  221 ;  of 
four  terms,  223  ;  of  illicit  middle, 
22-4 ;  of  illicit  major,  224  ;  of  illicit 
minor,  224  ;  of  particular  premises, 
225  ;  of  negative  premises,  225 ;  of 
equivocation,  22G,  22S  ;  of  composi- 
tion and  division.  228 ;  of  accident. 
231  ;  of  presumption,  240  ;  of  petitio 
principii,  240  ;  of  non  sequitur,  253  ; 


INDEX 


401 


of  false   cause,    255 ;    confusion  of, 

256-261  ;  inductive,  363 
Fallacy,  definition  of,  219 
False  cause,  fallacy  of,  255 
False  consequent,  253 
False  propositions,  144 
Figures  of  the  syllogism,  183 
Form  and  matter,  9,  16 
Formal  fallacies,  223 
Formal  logic,  1,  12,  15 
Fowler,  79,  81,  318,  321,  324,  364 
Fundamentum  divisionis,  95 

General  terms,  34-36 ;  mathemati- 
cal generals,  25,  75,  236  ;  logical  gen- 
erals, 25,  101,  236;  propositions,  110 

Generalization  of  terms,  57  ;  of  obser- 
vations, 21  m; 

Genus,  82,  86,  88  ;  mathematical,  93  ; 
logical,  93  ;  contrasted  with  differen- 
tia, 91 ,  92 ;  summum,  90 

Grammatical  division  of  propositions, 
107 

Hamilton,  1,  4,  73,  95,  154,  262,  266, 
267,  270,  271,  275,  285,  291,  338 

Hypothesis,  343 

Hypothetical  propositions,  107 ;  rea- 
soning, 204-211 

Identity,  law  of,  145 

Ignoratio  elenchi,  149,  153,  241-253 

Illicit  process  of  major  term,  1 74,  177, 
224 ;  of  middle  term,  174,  177,  224  ; 
of  minor  term,  174,  178,  224 

Immediate  inference,  154-170 

Imperfect  induction,  295 

Incomplete  disjunction,  214 ;  syllo- 
gisms, 197 

Indefinite  propositions,  110  ;  quantity, 
267 

Indirect  reduction,  193 

Individual  terms,  20,  32 

Induction,  nature  of,  295-313 ;  perfect, 
295,  296,  300-302,  320,  327;  princi- 
ples of,  328  ;  fallacies  of,  363-366 

Inductive  logic,  15,  295 ;  method,  298, 
340,  353;  inference,  313,  321-323; 
syllogism,  form  of,  306 


Inference,  nature  of,  154  ;  immediate, 
155  ;  mediate,  154,  171-189 ;  by 
privative  conception,  160 ;  by  con- 
tribution, 168;  deductive,  15,  279, 
295,  327,  328;  inductive,  313,  321- 
325. 

Infima  species,  90 

Infinitated  conceptions,  165 

Intension  of  terms  or  concepts,  68- 
81  ;  relation  between,  and  exten- 
sion, 72  ;  law  of,  and  extension,  73 

Introspection,  342 

Inversion,  156 

Inverted  propositions,  115 

Irrelevant  conclusions,  245 

Jevons,  37,  50,  63,  64,  90,  96, 103,  159, 
162,  193,  213,  220,  232,  239,  244,  263, 
264  note,  265  note,  280,  312,  348 
356 

Johnson,  Dr.,  250 

Joint  method  of  agreement  and  dif- 
ference, 358 

Judgment,  5, 16,  28-29 ;  intensive  and 
extensive,  123 

Karrtyopeojievov,  106 

Keynes,  13,  32,  37,  38,  79,  81,  187, 193, 
213,  270 

Language,  27 

Law,  conception   of,  6-9  ;  of  thought, 
6-9;    of    intension    and   extension, 
73;    of  contradiction,   145,  291;  of 
excluded  middle,  145,  292 ;  of  iden- 
tity, 145,  291  ;  of  sufficient  reason, 
292,     329 ;    of    universal   causation, 
318,  329 ;  of  the  uniformity  of  nat- 
ure, 318,  329,  331 
Laws  of  thought,  6-9,  290-294 
Limitation,  conversion  by,  156,  157 
Logic,     definition  of,     1 ;    relation   to 
the  other   sciences,  12  ;  divisions  of, 
15;  inductive  and  deductive,  15 
Logical  concepts,  26 ;  276,  277 
Logical  definition,    101  ;  generals,  75, 

101,  236 
Logical  doctrine,  formal  elements  of, 
18 


402 


INDEX 


Major  terra,  171 ;  illicit  process  of, 
174,  224 

Mal-observation,  364 

"Margarita  Philosophica,"  233 

Matter,  form  and,  9 

Material  fallacies,  225,  238-261 

Mathematical  concepts,  25,  26,  276, 
277 

Mathematical  generals,  75,  236 ;  rea- 
soning, 275-289 

Mediate  reasoning,  171-189 

Method,  scientific,  800;  inductive, 
298,  340,  353;  deductive,  337;  of 
agreement,  297,  354  ;  of  difference, 
297,  355 ;  of  concomitant  varia- 
tions, 357;  of  joint  agreement  and 
difference,  358  ;  of  residues,  358 

Middle  term,  171,  174-180,  224 

Mill,  J.  S.,  37,  41,  70,  79,  80,  81,  203, 
247,  250,  317,  355 

Minor  term,  171 ;  illicit  process  of, 
174-180,  :2'.'4 

Mnemonic  lines,  190 

Miscellaneous  propositions,  114 

Modal  propositions,  114 

Modus  ponendo  tollens,  216 

Modus  ponens,  205 

Modus  tollendo  ponens,  216 

Modus  tollens,  205 

Moods  of  the  syllogism,  1S1-189 

Negation,  conversion  by,  159 
Negative  propositions,  109 ;  distribu- 
tion of,  140 
Negative  terms,  44 ;  signs  of,  45 
Nego-positive  terms,  44-46 
Newton,  304,  350,-300 
Non  causa  pro  causa,  255 
Non-observation,  363 
Non  sequitur,  253,  256 
Notion,  21 

Observation,  342,  351 ;  errors  of,  363 

Obversion,  160 

Opposition,   141-153  ;    application    of 

the  principles  of,  144  ;  rules  for,  144  ; 

square  of,  144 
Or,  signification  of,  109,  212 


I'\kticular  propositions,  109,  110; 
signs  of,  112 

Partition,  99 

Partitive  propositions,  116 

Per  accidens,  conversio,  156 

Percepts,  .':.! 

Petitio  argumenti,  241 

Petitio  principii,  fallacy  of,  240-245 

Plurative  propositions,  110 

Porphyry,  tree  of,  97 

Positive  terms,  44 

Post  hoc,  ergo  propter  hoc  fallacy,  255 

Predicables,  82-94 

Predicate,  106;  distribution  of,  138; 
nlation  between  subject  and,  122- 
140.  See  also  Quantification  of  pred- 
icate 

Preindesignate  terms,  267 

Premises,  29,  171 

Primary  laws  of  thought,  290 

Privative  conception,  44 

Probation,  338 

Proof.     See  Probation 

Proper  names,  31-33 

Property,  S2 ;  kinds  of,  82-85 ;  table, 
88  ;  propositions,  105-121 ;  defined, 
17  ;  divisions  of,  197,  121 

Prosyllogism,  198 

Psychology,  13 

Pure  propositions,  114,  277,  310-312 

Qualitative  reasoning,  277,  310-312 
Quality  of  propositions,  109 
Quantification  of  the  predicate,  262- 

274 
Quantity  of  terms.    See  Extension  ;  of 

propositions,  109 
Quanto-qualitative  reasoning,  277 
Quaternio  terminorum,  223 

Reason,  law  of  sufficient,  329 
Reasoning,  5,  16,  29-30,  171-180; 
forms  of,  197-203 ;  hypothetical, 
204-211 ;  disjunctive,  212-218 ;  math- 
ematical and  other,  275-2S9  ;  quan- 
titative, 277  ;  qualitative,  ^77 
Reduction  of  the  moods  and  figures, 
190-196 ;  of  hypothetical  syllogisms 
to  categorical,  209 


INDEX 


403 


Reisch,  233 
Relative  terms,  47^49 
Residues,  method  of,  358 
Rules  of  the  syllogism,  173 

Science  and  Art,  1-2 

Sciences,  mathematical  and  metaphys- 
ical, 35 

Scientific  method,  300-366 

Secondary  laws  of  thought,  293 

Simple  accident,  fallacy  of,  231 

Simple  conversion,  158 

Singular  propositions,  110 

Singular  terrns,  31-33 

Socrates,  296 

Sorites,  201 

Specialization,  59-63 

Species,  82,  88 ;  contrasted  with  genus, 
91-92;  and  fallacy  of  accident,  231, 
235 

Specific  accident,  fallacy  of,  231 

Subaltern  propositions,  143 

Subalternans,  143 

Subalternate,  143 

Subcontrary  propositions,  143 

Subject  and  predicate,  106 ;  distribu- 
tion of,  138  ;  relation  between,  122- 
140 

Subordinate  species,  96 

Sufficient  reason,  law  of,  329 

Summum  genus,  90 

Superordinate  conceptions,  96 


Syllogism,  IS,  171-180 ;  forms  of,  197- 
203;  hypothetical,  204-211  ;  disjunc- 
tive, 212-218;  inductive,  306 

Syncategorematic  terms,  31 

Synthesis  of  percepts,  20 

Synthetic  propositions,  112 

Tautologous  propositions,  114 
Terms,  31-49 ;    ambiguity   of,  50-67 ; 

intension  and  extension  of,   68-81  ; 

denotation  and  connotation   of,  79  ; 

distribution  of,  138 
Terminorum,  quaternio,  223 
Theory,  344 
Thompson,  1,  266 
Thought,  definition  of,  3-6;  nature  of 

its  laws,  0-9  ;  laws  of,  290-294 
Traduction,  279,  310,  312 
Tree  of  porphyry,  97 
Truistic  propositions,  114 

Ueberweg,  1,  134 

Universal      propositions,     109,      110 ; 

signs  of,  111  ;  distribution  of,  140 
Univocal  terms,  51 

vnoKtiiJiCvov,  106 
Verification,  341,  349 

Watts,  Dr.,  1 
Whately,  1,  233,  245,  247 
Weakened  conclusion,  184 


THE  END. 


CENTRAL  UNIVERSITY  LIBRARY 
University  of  California,  San  Diego 

DATE  DUE 

NOVO7  1980 

a  39 

UCSD  Libr. 

Wii 


VAV.V.'.v 


